$B    ST    t,li3 


THE  LIBRARY 

OF 

THE  UNIVERSITY 

OF  CALIFORNIA 

EDUC-PSYOI 

GIFT  OF 


Emily  Turner 


'"  ANSWERS 


Wentworth's  Algebra, 


vi?Mm»iL*i 


tTJ^JKIWiVi 


-^ 


EDUC- 
PSYCH. 
LIBRARY 


(pA/Sd 

h/W 

^^Si^ 

d^^^^^p" 

ANSWERS. 

t^C. 

B^     • 

Exercise  I. 

y(lf 

1.   22.           6.   94.          11.   6.            la 

25.           21.   39|. 

2.   26.            7.    81.          12.    1.             17. 

64f.         22.   20t 

3.   564.          8.    16.          13.    120.        18. 

31.           23.    20|. 

4.    6.              9.    2.            14.    25.          19. 

-15.        24.   0. 

5.   39.          10.    127.        15.   43.          20. 

24.           25.    lOJ. 

26.   a  +  h.         27.    2x.         28.    a -5. 

29.   x^\. 

30.   x-2,  x-\  X,  x-\-\  x-\-1. 

32.    y. 

31.    ax. 

33.    a -5. 

34.   ah-^c.            35.    ^~^^=a5m- 
c 

-6.+    I 

36.      ^^     +6b(c  +  d-a).                 37. 
m  —  a 

4a^-5a2^,2_^35*. 

Exercise  II. 

3.  30,  5.  5.    60  ft.  broken  off;  15  ft.  standing. 

4.  25  cents.  6.    27,  36. 

7.  Horse,  $168;  cow,  $42;  sheep,  1 6. 

8.  Apples,  13  cents ;  pears,  26  cents ;  oranges,  52  cents. 

9.  Harness,  $50;  wagon,  $100;  horse,  $200. 

10.  Thomas,  $1.20;  Richard,  $1.20;  Henry,  60  cents. 

11.  A,  $100;  B,  $400;  C,  $500. 

Exercise  III. 

1.   5.  3.   -11.  5.   51.  7.   $2484. 

40.  4.   -3.  6.   495.  8.    B.C.  241. 

D.  14.  10.    150  steps  in  all ;  76  steps  forward. 

077 


ALGEBRA. 


Exercise  IV. 


1.  0.  4.   3a;2.  7.  25  my.  10.    14  aV. 

2.  6mx.  5.    7my2.  8.  -22aJ2.       11.    Qb^m^ 

3.  —  20ww^.     6.   2  ah.  9.  —  55^y.        12.    9  a  — 106. 

13.   —a^c-\-xi/z.  14.   3a;^y  —  4a6  — 2mn. 

Exercise  V. 

1.  9a  +  9h  +  9c.      3.   2a  +  2^».      5.    -2a+2^>+2d 

2.  a-b.  4,   a  +  b  +  c.     6.   2a^-9ar^~8x+10. 
7.   ux^+4:a^-i-Sx^^2x-9.  9,  6ab-{-7ax'-9a^xi-aa^. 

IL   -2:ry-23/2+ll2/z  +  52?-6y-62;. 

12.   6m'  —  m^n  -}-  2mV  +  3 mV  —  4 mw*  +  5 w*. 

Exercise  VI. 

1.   9.       2.   -25.        3.   -89.        4.   200.       5.   506  years. 
6.    35°  in  latitude  ;  75°  in  longitude. 

Exercise  VII. 

1.  9x.  6.  Alaa^.  11.  6a^. 

2.  -8  ah.  7.  8  A.  12.  13 a;^^. 

3.  -1ab\  8.  a;y.  13.  -6a^. 

4.  22?nV.  9.  8a:p  +  3ay.  14.  -3a5  +  6ma7. 

5.  -4ay.  10.  2ab^y-aby.  15.  3a  — 25  + 4c. 

Exercise  VIII. 

1.   4«  +  2c.  2.   a +  65  + 4c.  3.    2x^-2x-^. 

4.  3a;*-a;3-14a;  +  18.       6.    -5:ry-5^z+y2+2y2 

5.  T?-ax-^2a^.  7.   2a«-6a25  +  6a52_253. 

8.  —\xy-\-2xz  —  'i^-\-hyz  —  '^. 

9.  a^+7a5:r-5a:2  4.j2^_|.125^  +  :rl 

10.   -2r^;«  +  3a;y2_2/3_i4^_|_2^3/_10y2-|-6. 
11,   -a^-8a5^  +  5^  12,   -  2a;^  +  4a;''y-8a7?/^-8/, 


ANSWERS. 


13.   a'-W  -Za%c  -ZaWc-  oh(?  -  a^(?  -W(?. 
14.    2a  +  46-9c  +  (^.         15.    h-a\  a^  ^  a^h -\- ^  ab^ -\- ¥ . 

16.  3?-^xy-\-'l^.  la   -10a2-4a5  +  c2. 

17.  19ac+17cc?-17.  19.   xy  -"Ix  —  ^y -\-\, 

20.   -2a26c-2a52tf  +  2a5r-2a5c.         21.   52^^-50;  + 5. 

22.  ^  +  a:2/  +  3/2.  24.    2hx-\-2hy. 

23.  a2(?--hf-cc(?-\-dy\       25.    1x-ly  +  2f. 

26.  a262+12a5c-9aa;2_4^j2_|_g^^^_3^2^_ 

27.  -a2-652_|.^  29.   rr^-Sy^-Szl 

28.  2a2.  30.    2a3_2a2c  +  2ac2. 


Exercise  IX. 

1^ 

25. 

7.   a  +  a^ 

13. 

-5  + 10c. 

2 

a-{-h-2c. 

8.   2a  +  3t?. 

14. 

—  5  a. 

3. 

Zx-^y-z. 

9.    3a-35. 

15. 

4a-166-2c 

4. 

a~h-\-  c. 

10.   2a- h. 

16. 

3a-2c. 

5. 

-2x. 

11.   5  a. 

17. 

16  -12:r. 

6. 

-96-f  14c. 

12.   4a;  — y  — z. 

18. 

4.C. 

19. 

3  a. 

Exercise  X. 

20.    a. 

1.  (2a-35)-(4c-c?)  +  (3e-2/); 
(2a-36-4c)  +  (c?+3c-2/). 

2.  (a-2a;)  +  (4y-32;)-(25-c); 
(a-22;  +  4y)-(3z  +  26-c?). 

3.  (a^  +  3a'')-(2a3  +  4a2)+(a-l); 
(a«  +  3a*-2a3)-(4a2-a+l). 

4.  -(3a  +  26)  +  (2c-5c?)-(e  +  2/); 
-(3a  +  25-2<?)-(5c^+e  +  2/). 

5.  (ax  —  hy)  —  {cz  -f  5a;)  +  (<?y  +  az)  ; 
(aa;  —  hy  —  cz)  —  (bx  —  cy  —  az). 

6.  (2x'  -Sx^y)-{-(4::t^y'  -  5x'f)-i-(xy*  -  2y^) 
(2^^  -  dx*y  +  4:x^y')-{5x'f  -  xy*  +  2y'). 


ALGEBRA. 


'1.   (2a-35-4c?)  +  (cZ+[3c-2/]). 

2.  (a-2x  +  4:y)-(Sz-\-[2b--c]). 

3.  (a'  +  Sa*-2a^)-(4.a^-[a-lJ). 

4.  -(Sa  +  2b-2c)-(bd+[e-}-2f]). 

5.  {ax  —  hy  —  cz)  —  {hx  —  \_cy  +  as:]). 

6.  (2s(^-^x^y  +  ^:^y')-{bxY-[xy^~2'i/]). 

8.  (2a-4Z>-2c)a:-(6a  +  3c)y  +  (4^)z. 

9.  (a-Q:r  +  (2a  +  3)2/  +  (4a-35-2)0. 

10.  la-U-2c)x-{2b-^2>c-\-a)y^{a^U-\-bc)z. 

11.  (12a-15c):r  +  (12a  +  45  +  6c)y-(125-3c)z. 

12.  l2a-2h)x-{Zh-\-c)y. 


1.  - 136. 

2.  -  320. 

3.  - 18.0621. 

4.  90. 


9. 
10. 
11. 
12. 
13. 
14. 


11. 


-12al 
45  m^n. 

—  \2ahxy. 

—  8  cdmn. 


Exercise  XI. 

5.  -258. 

6.  45,700. 

7.  61,225. 

8.  82.477956. 

Exercise  XII. 

15.  -10a'"+". 

16.  21  aV. 

17.  22^abc. 
18. 
19. 
20. 


-96a2^V. 

-  18,954  a^^m/. 

-md^hSJ. 

Exercise  XIII. 


9.  -  728. 

10.  2052. 

11.  183.92. 

12.  -1718.382762. 

13.  -8.0001703416. 


21. 
22. 
23. 
24. 
25. 
26. 


-  42  rn^qo^. 
- 144/^«. 
24a«mV. 
162a;VV. 

-  24  ^2^,2^,  V. 

-  168  a*5«mW 


\2(^h-^ah^.  7.  6;rV-8a:2^  +  10ar^yz^ 

24a*-27a3^.  8.  aVy- 5  A^ -faVy  +  2aa;V 

9.  27a«6^  _  9  a^js  +  12  aW  +  3  ah^. 
10.  -15a;»y  +  10^y  +  35:ry-5a;2^^ 
12^:32^ -15a7y- 24 r'y.  12.  ^a^-2a^b-a%\ 

13.  Sa^^y^H^^V-lSa^^y^z^+lSa^y^-Oa^yzl 


ANSWERS. 


Exercise  XIV. 

1.  x*-[-s?~20.  4.  a?-f.  7.  x^  —  2x-\-l. 

2.  3/^+7y-78.         b.  2x'-{-2>x^j-2f.      %.a?-<^a^x. 

3.  a^-x\  6.  6  ^'^-96. 

9.  -  10  5^  -  aZ)2  -f  26  a^^  -  Ya^. 

10.  2o^  +  bah  +  2h\  13.  6a35-23a252  +  20a53. 

11.  a«-Z>^  14.  --^a'-^aW+l^aW-Qab\ 

12.  a^  +  ^^     '  15.  a^  +  a^h^  +  h\ 

16.  a«  -  5  a*^  +  10  a^h^  -  10  aW  +  5  a5*  ~  5^ 

17.  :r2  -  43/2 +12y2;- 9^1 

18.  ^x^  +  x^y~l^xif-\-2a^yz-]-2,xyh-\-^fz. 

19.  :r^  4-  0!^y  +  n^y  +  ^2;  +  ^3/z  +  ^y^^  +  ^^^^  +  ^y^^  +  2/^2;^- 

20.  a2-3a5c  +  ^^  +  ^^  23.  ar'-41a;-120. 

21.  a;2  +  3:ry +  3/3-1.  24.  :rH  10^7-33. 

22.  ^+151a:-264.     25.  x'-lx^^21x'-llx''-2b3?-^^a?-{-2x. 

26.  ^o^~bo(^-\-^x^-l03?-^x'-bx-^. 

27.  25  a^Z)  -  9  a'¥  +  22  a^b''  -  4  a^^^  +  aV. 

28.  4  a^y  +  8  aV  -  ^4  aV  -  128  a^f. 

29.  6  7)1^71  —  IS  m^n^  -[-18 7?i3n^  —  6  mn^. 

30.  24a«^>-26a^^^^+4a*^)«-6aW-5a258-f4a6»-35i». 
31.  a;*-10^  +  9.         32.  a^  +  x^+1.         33.  a^  +  a*b*  +  b\ 

34.  32  a^5  -  8  a^b'^  -f  8  a^^^  -  6  a*Z>*  -  2  a^b^  -  a%^  +  a6^ 

35.  a:^  +  ax"^  -  27aV  _  13  ^3^  _^  134  ^4^  ^  120  a''. 

36.  531,441  oP-  -  h'^  37.  -  ?/ V  -  3  2*. 

38.  ^a'b-o^b^-^aW-^a'b^+2a¥-2ab^+U*-&b\ 

39.  _  «4  _|_  2c,2J2  _  J4  _|.  2  aV  +  2Z»V  -  c^     40.  J2  _  d\ 

41.  4a2+4?/-F4c2+4c?2  46.  2/r.  51.  -6:r. 

42.  2^2  +  252 -f  2^2.  47.  0.  52.  25a*- 25^^ 

43.  ac-(?.  48.  0.  53.  24a5 -12052. 

44.  2mn.  49.  0.  54.  ^x^y^-\-2o[?f—f. 

45.  2aH25'  +  2c2.  50.  -3Z>a:.  55.  0. 


ALGEBRA. 


Exercise  XV. 

7.  x'^^xy^f.    9.  4^2^4;r+l.         11.  \-'lo?-\-x^. 

8.  /-2y2+22.    10.  4a2-f20a5+2552.  12.  9aV-24«:r«+16:r''. 

13.  l-14a  +  49a2.  17.  144  + 120^ +  25 a;^. 

14.  25a:2/+20a:y+4.  18.  l^x'y'' -'^x^z^ -\-fz^. 

15.  a2^,2_|_2aZ,cc^+^c?2  19.  9aW- 6a5V^+5V(^2 

16.  9m^n^—24:mn-j-16.  20.  16^;^- 8:ry +  :r'y*. 

21.  ^t^-f.  24.  9 a2Z,2 _ 4 J4  27.  36 x'f  -  25 y^ 

22.  4:a^-b^  25.  16:r^-92/^  28.  16:r^«-l. 

23.  9-ar^.  26.  aV-^y.  29.  l-9a2Z)6. 

30.  aV-Z>y. 

Exercise  XVI. 

1.  a^  +  fi-z^  +  2xy  +  2xz  +  2yz. 

2.  a;2  +  3/2_|_22_2<^y_|_2a;2-22/0. 

3.  m^-{~n^-{-p^-{-q^-]-2mn—2mp--2mq—2np—2nq-{-2pq. 

4.  a;*  +  4:^  +  9  +  4a;«-6a;2_i2:r. 

5.  a;^  +  36:r2  +  49-12a;3  +  14a;2-84:r. 

6.  4a;^  + 49:^2  + 81 -28a:3  +  36a;2_i26rr. 

7.  a;4_|_y4_j_24_j_2a;2^-2a:2^-2i/2z2. 

8.  .i;8  +  18a:y  +  2/«-8^y-8^/. 

9.  «6_|_j6_^^6_^2a«^«  +  2aV+25V. 

10.  ^«  +  2/«  +  2«-22;«2/«-2a;«2«  +  22/«^. 

11.  x^-j-Af  +  9z^  +  4:xy-&xz-12yz. 

12.  ,;*  +  4y^  +  25 .-^  -  4  rr^y^  +  lOr^.^  -  20y2^. 

13.  ^4^4_|_4^_8a;.         14.  x'  +  Sdx'  +  ^Q-lOa^-IOx. 

15.  4a;^-7.r2+16-12:r3  +  24^. 

16.  ^  +  42/2_|_9^2_^4^y_|_6^2;  +  122/2:. 


Exercise  XVII. 

1.  ^  +  5a;+6.  3.  a:^- 9a;+ 18.  5.a^-1x-S. 

2.  ^+6a:+5.  4.  :r2-9:r  +  8.  6.  ar*  +  3 a: -  10. 


ANSWERS. 


7.  ar'  +  4:x-21. 

8.  x^-6x  +  8. 

9.  .T2+12r^-+ll. 

10.  Q!^-\-ax  —  6a^. 

11.  x^  —  (c  +  d)xi-cd. 

12.  X^—Sx7/—4:f. 

13.  a2_7j^4_io52. 


14.  x*  +  Sa^f-\-27/. 

15.  a;'^-2:r3y-3rr22/2^ 

16.  a^x^  —  Sax  —  54:. 

17.  rr^  +  (a  —  Z))  :i;  —  a5. 

18.  x^  —  7x  —  4:4:. 

19.  :r2  +  ^-132. 

20.  :^- 15:^  +  50. 


o^+Sx^a+Sxa'+a^ 
a^~?>x'a+^xo?-a^ 

0(^+^X^+^X-\-l. 

a^-'^x^^^x-l. 


Exercise  XVIII. 

5.  x^-{'^c(?a+(Sx^a^-\-4:xa^+a\ 

6.  x'^-A:x^a+(ja^o?-4:xa?-\-a^. 

7.  rr*  +  4^  +  6n^  +  4:i'  +  l. 

8.  rr^-4^3_|_g^,2_4^_|_]^_ 

9.  x^-{-^oxSj  +  lOa^f-\-lO:x^f  +  bxy^-\-y^. 

10.  .7;^  —  5a;^?/  +  lOa^if  —  lOx^  +  bxi/~ if. 

11.  .^'^  +  5:r^  + 10^.^+10:^2  +  5:^  +  1. 

12.  :r^  -  5:?;^+10:r«- 10:^2  + 5a; -1. 


Exercise  XIX. 
1.    66.  4.   -214.  7.   -11.  10.   -.022^. 


2. 

581. 

5. 

-21.7. 

8. 

43^. 

11. 

.3183+ 

3. 

-128. 

6. 

-1.23. 

9. 

-  .1123. 

12. 

.0101321+. 

Exercise  XX. 

5. 

Bm. 

11. 

X 
57/' 

17. 

9m 

y 

23. 

8abh\ 

6. 

-4a^ 

12. 

-9a^ 

18. 

24. 

-  UxYz^. 

7. 

5a 

13. 

-35m 
4  a:?; 

19. 

4:mx* 
5a« 

25. 

8b:^. 

8. 

1 

6  a. 

14. 

h(?. 

20. 

Gx'z. 

26. 

o?bx. 

9. 

15. 

mV. 

21. 

UcV 

0? 

27. 

I1a\ 

10. 

7ac. 

16. 

-  17a?/. 

22. 

^am^n 

28. 

1a^-\ 

v<t 


ALGEBRA. 


Exercise  XXI. 

3.  —2am-{-Sbn  —  4:cp.  9.  -4:xf-{-6ar^7/  +  8c^. 

4.  —  7a  +  65-5d?.  10.  a;y-2^  +  3;ry-l. 

5.  Sx*-2a^  +  l.  11.  S-2ab-a'b\ 

6.  x-2a^  +  Sa^-4:X^.  12.' ~z-2xz^+5x^yz' 

7.  -5m2-4wy  +  22/2  ^g^  4ac3- 2aV  +  3a35c. 

8.  2a2_3a5  +  652. 


Exercise  XXII. 

6.  x-4:.  8.  a;2  +  a;  +  3.  10.  Sa^  +  2x+l. 

7.  a:  — 8.  9.  3^-2a:  +  4.  11.  a^  —  Sx  +  7. 

12.  r'  +  aj^  +  ^  +  ^  +  rr+l.     14.  a^  +  3x^i/  +  9xf  +  21f. 

13.  a2_|_«5_^2  15^  ^4_|_^^_^^^_|_^^_|_^4_ 

16.  a^-2a35  +  4a2^2-8aS3+165*. 

17.  2a2-6a25  +  18aS2_27^3 

18.  a;2_2a;  +  2.       20.  a;2_5^_^6        22.  n-^-^r-lQ. 

19.  xr^-Sx-1.       21.  :r2-4a;  +  8.       23.   l-3:r+2^-r^ 

24.  x^+2a^+Sx'+2x+l.  37.  4a;2^2^y  +  2/2^ 

25.  a2  4-2a5  +  352.  38.  -  8a^  + 2^26  -  aJl 

26.  2a^-^a^  +  2x.  39.  l+:r-2:r2. 

27.  a*+3a3+9  a2+27a+81.  40.  -  1  -  3  aJ  -  13  a^b\ 

28.  6a^-7x-\-S.  41.  :r^  +  2a^?/  +  23;y  +  ^3/^. 

29.  a:2_3a;-y.  42.  x^-Sxr^y+Sxf-1^. 

30.  o?-Zxy-2y'.  43.  a*  +  a^^^,^  3  Z)*. 

31.  a:^  +  ^y  +  2^.  44.  27a:2y -Y^xf-'^f. 

32.  a^  +  xy  —  f.  45.  a2_}_2a5  +  4^l 

33.  :z:-3/-2.  46.  a:^  +  3 ^r^y  +  8 a:^?/^ -  8 y^ 

34.  4-6a:+8a;2_i0:c3.  47.  ^aj"  ■\- ^ab +  U\ 

35.  rr  +  y.  48.  a^-\-b'^-\-(?  —  ab  —  ac—bc. 

36.  2^  —  Zxy  —  if.  49.  a  +  c  +  25. 

50.  a2  +  2a6  +  ^»2_^^_j^_|_^^ 


ANSWERS. 


Exercise  XXIII. 


1.  (b-\-c)a  —  he.  5.  3/^  —  (m  +  ^)  3/  +  '^'^^ 

2.  X  —  c.  6.  (x~{-  a)  x~  h. 

3.  :f^-{a-{-h)x  +  ah.  7.  x^ -~{h  + d)  x  +  hd. 

4.  {x-\-a)x  +  h.  8.  {x -\- c)  X -{- d. 

9.  a:*  +  o;^  +  :r^  +  .^'  +  1  —  wo;  (a:^^  +  a;  +  1)  +  no(^. 
10.  a;  +  y  +  2. 

Exercise  XXIV. 

1.  y2_j_y^i.  4.  .^.2_f_7^_^49^ 

2.  52+55  +  25.  5.  x^  +  a?y-\-a?y^-^xf  +  '/. 

3.  a2  +  6a  +  36.  6.  a^  +  a^^  a2  +  a+ 1. 

7.  l  +  2a;  +  42:2_ 

8.  x^'+^a^b  -{-  4:a^b^ -{-^xW  ^lQ>h\ 

9.  4aV  +  2aa;  +  l.  11.  16a252+ 12a5a;  +  9:r2. 

10.  l  +  3^y+9:ry.  12.  81a*+27a3+9a2+3a+i. 

13.  16a*  +  24a35  +  36a252_|_54^j3_^815*. 

Exercise  XXV. 

1.  x^  —  xy^f.  8.  64^?/2__8^^^_|.^_ 

2.  x^-a^y^a^f-xf+^f.  9.  Sla^- 54a5  +  36^2. 

3.  l-2a  +  4al  10.  16^2-40^5  +  10051 

4.  9a2-3a5  +  52.  11.  16a252- 12a5a;  +  9:r2. 

5.  4aV-2a:r+l.  12.  x^-1x-\-4c^. 

6.  a:2_3^^y_j_9y2  13^  9a;2^-6^y2  +  4z2. 

7.  a^-2a35  +  4a252-8a53+165^ 

14.  256  a"  -  192  o?b  +  144  ^252  -  108  ab^  +  81  b'. 

Exercise  XXVI. 

1.  a^  +  x^y-^xf  +  f.  6.  x^-^x'y +  ^xf~21f. 

2.  a^-a^y  +  xf-f.  7.  8;r«  +  4^2_|_2:i;+ 1. 

3.  a^+a*.2;+aV+a2:f2+a:?;*+a;^     8.  ^a? -4:0^ +  2x ~l. 

4.  a«-a^a;+aV-a2:r3+aa;*-ar^.     9.  27aV  +  9a2:r2+3aa;+l. 

5.  x^^2>a^y  +  ^xf  +  21f.      10.  27aV-9a2a;2^3^^_l 


10  ALGEBRA. 


11.  32a*  +  Wa'b  +  Sa^b^  +  4.a'b^  +  2ab'  +  b'. 

12.  S2a' -  16a'b  +  8a'b^ -  4:a'b^  +  2ab' -  b'. 
is!  a^  +  Sx*i/  +  9a^i/  +  21a^f  +  81x/  +  2ASf. 

14.  c(^ -Sx^i/ -^9a^f  -  27T^f  +  8lxy^ ~24S7/. 

15.  27a«  +  18  a^c  +  12  ac^  +  8  c-^. 

16.  27 a^  -  18  a^c  +12a(^~8  c^ 

17.  64a«  +  160a2+ 400  a +1000. 

18.  64a»-160a2  +  400a-1000. 

19.  125a;3  +  25a;2  +  5^+l. 


Exercise  XXVII. 

1. 

rr^-^y^  +  y*. 

6.  a^-x* 

+  1. 

2. 

a^-a^+l. 

7.  16:^;^- 

Axy+y\ 

.3. 

a^  — aV+a 

Y~ay+f. 

8.  16-4 

a'  +  a'. 

4. 

b'-b'  +  b' 

-b'+l. 

9.  81a*- 

9a'b'  +  b\ 

5. 

a^-a'b'  +  b'. 

10.  81c*- 

9c2+l. 

Exercise  XXVIII. 

1. 

X  =  4:. 

7.  x  =  6. 

13.  x  =  —7. 

19.     X  =  4:. 

2. 

x  =  2. 

8.  x--=9. 

14.  :r  =  3. 

20.  X  =  10. 

3. 

X=:2. 

9.  x=7. 

15.  x  =  7. 

21.  a;  =  3. 

4 

x  =  5. 

10.  a;  =  11. 

16.  x  =  0. 

22.  X  =  15. 

5. 

x  =  l. 

11.  x  =  9. 

17.  x  =  8. 

23.  :r  =  2. 

6. 

x^l. 

12.  x  =  2. 

18.  x  =  6. 

24.  x  =  S. 

25.  :?;  =  4. 

Exercise  XXIX. 

6. 

70.       7.  43.       8.  23. 

9.  21,  7.        10. 

36,  26,  18,  12. 

11. 

8,12. 

14.  $68. 

12. 

Father,  50 

yrs. ;  son,  30 

yrs.       15.  Cloth 

,$3;  silk,  $6. 

13.  30,  26,  22,  18,  14,  10.  16.  52. 

17.  A,  $130;  B,  $150;  C,  $130;  D,  $90. 

18.  152  men ;  76  women  ;  38  children. 

19.  13,  21.  20.  A,  84  yrs. ;  B,  26  yrs. 


ANSWERS.  11 


21.  Wife,  14000;  each  son,  $1000;  each  daughter,  $500. 

22.  49  gallons.      24.  50,  40.      26.  24,  60. 

23.  $21.  25.  42,  18.      27.  A,  88  years;  B,  44 years. 

28.  18  years.  30.  A,  57  years;  B,  19  years. 

29.  40  years.  31.  4  years. 

32.  10  dollar  pieces ;  40  twenty-five-cent  pieces. 

33.  28  pounds  of  better  kind ;  2  pounds  of  poorer  kind. 

34.  28  days.  35.  60  gallons  @  $2 ;  40  gallons  @  $3. 

36.  40  children.         37.  31 J  gallons.         38.  $45. 
39.  2  half-dollars;    6  quarter-dollars;    24  dimes;  48  half- 
dimes.  40.  10  half-dollars ;  41  quarter-dollars. 
41.  80  guineas ;  128  half-crowns. 

Exercise  XXX. 

1.  5a(a-3).  6.  ?>a%^  {^a% -la-^-^h^). 

2.  6a(a2-f  3a-2).  7.  21xY{2-{-4.x'f -^xY). 

3.  l{lx'~^x  +  2).  8.  45^y(:iV-2^-82/). 

4.  ^xy{x^-2>xy-\-2y"-)  9.  10ay\o?-2ay+?>f). 

5.  y{f-af-^by  +  c).  10.  ^2o:'h\l  +  ^aW -A:a^¥) 

Exercise  XXXI. 

1.  {x  —  a)  {x  —  h).  6.  {ab-{-pq){x  —  y). 

2.  (a  —  y){b-^  y).  7,  (ex  -f  ay)  idx  —  by). 

3.  {b  —  x)(c-\-  x).  8.  (ac  —  bd)  (by  —  dx). 

4.  (m -{-a)  (x-\- n).  9.  (a  —  b)(x  —  y). 

5.  (cx  +  y)(dx~y).  10.  (cz  +  y)  (dz  —  y). 

Exercise  XXXII. 

1.  (x  +  S)(x  +  3).  5.  (x  +  11)  (x  -f  10). 

2.  (x  +  6)  (:^  -f  5).  6.  (y  -f  20)  (y  +  15). 

3.  (2/-M2)(2/  +  5).  7.  (b  +  11)(b  +  6). 

4.  (z  +  12)(z  +  l).  a  (x  +  2)(x  +  l). 


12 


ALGEBRA. 


9.  (x  +  6)(x  +  l). 

10.  (a  +  8b)(a  +  b). 

11.  (x  +  9a)lx  +  4:a). 

12.  (y+16;?)(y  +  3^). 

13.  (z  +  25q){z  +  4:q). 

14.  (a2  +  3)(a2-f  2). 
16.  (^  +  3)(z«+l). 
16.  (a5  +  16)  (a^  +  2). 

25,  (rs  +  18 

26.  (mV  + 


17.  (a;y  +  4)(:^;y  +  3). 

18.  (2*  +  8)(0«  +  2). 

19.  (a  +  6^»)(a  +  45). 

20.  (a;3_|.5)(^^_|_4>^ 

21.  lax+llh)(ax  +  Sb). 

22.  (ac  +  5a:)(a(?  +  2:z:). 

23.  (xi/z+U)(x^z  +  S). 

24.  (^)c  +  13a)(^»c  +  5a). 

2)(?'S  +  5z). 
17jO^)(77lV+3^^). 


Exercise  XXXIII. 


1.  (x-5)(x~2). 
.2.  lx-19)(x-10). 

3.  (a -12)  (a -11). 

4.  (^  -  20)  (5  -  10). 

5.  (2 -23)  (2 -20). 

6.  {x-e)(x-l). 

7.  (x'-3a^)(x'-a'). 

8.  (a;-6)(a;-2). 

9.  (z- 56)  (2-1). 

10.  (2/3 -4)  (2/3 -3). 

11.  (^y-26)(a:y-l). 

12.  (a'b^-6)(a^b^~6). 


13.  (a5c-ll)(a^c-2). 

14.  (a:-10)(:r-5). 

15.  (:r-10)(a;-10). 

16.  (a^-18)(a:r-3). 

17.  (ax-lSb)(ax-Sb). 

18.  (ac-lSz)(ac  — 11  z). 

19.  (5;-13)(:^-7). 

20.  (x—15y(x  —  8). 

21.  (0-45)  (0-8). 

22.  (x  —  a)(x~  c). 

23.  (y2;  -  17ab)  {yz  -  11  ah). 

24.  (cc?-  17a5)  (c?c?-  13  a^). 


Exercise  XXXIV. 


1.  (a;  +  7)(a:-l). 

2.  (:^+12)(a:-7) 

3.  (y  +  12)(y-5) 

4.  (y+15)(y-3) 

5.  {z  +  12)  {z  -  1) 

6.  (2+20)(z-7) 

7.  (a +  25)  (a -12) 


8.  (a +  30)  (a -5). 

9.  (5^  +  4)(Z>^-l). 

10.  {be -\-li)  {be -11). 

11.  (c^  +  20)(c-^-5). 

12.  (d?  +  30)(c-13). 

13.  (a +  12)  (a -11). 

14.  {xyz^ll){xyz  —  2). 


ANSWERS.  13 


Exercise  XXXV. 


1.  (x  -1)(x  +  4).  9.  (y  -  10a)  (y  +  5a). 

2.  (y  -  9)  (2/  +  2).  10.  (ab  -  4)  (a^  +  1). 

3.  (x~12)(x  +  S).  11.  (a:c  -  9)  (ao;  +  6). 

4.  (2  -  15)  (0  +  4).  12.  (cc?  -  30)  (cd  -f  6). 

5.  (z  -  14)  (2  4-  1).  13.  (a^c  -  2)  (a^c  +  1). 

6.  (a  -  20)  (a  +  5).  14.  (/z^  _  12)  (yV  +  7). 

7.  (c^  -  10)  (c«  +  1).  15.  (ab  -  18)  (a5  +  2). 

8.  (:r-10)(a:  +  2).  16.  (x -a)  (x  +  b). 

Exercise  XXXVI. 

1.  (x  +  ey.  5.(2/4-100)1  9.(^+12)2. 

2.  (:r  +  14)2.  Q    (■^  ^  7)2_  10^  (^2  _^  gl)2^ 

3.  (:r  +  17)'.  7.  (:r  +  ISyf.  11.  (2  a  +  3  ^>2)2. 

4.  (z  +  1)1  8.  (y2  +  8  2^)2.  12.  (Sx2^  +  5  z)\ 

13.  (3:c  +  2y)2.  14.  {2a^x -\~ba^y)\ 

Exercise  XXXVII. 

1.  (a -4)2.  9.  (^3-17)2.  17.  (3mn-4)2. 

2.  (a  -  15)2.  10.  (2  x'y-b  fzf.  18.  (2  ^»a;  -  3  xyf. 

3.  (:r-19)2.  11.  (4:r3j2-y22>)2  j9^  (7a-8^.)2. 

4.  (a: -20)2.  12.  (Sabc-bcdf.  20.  (8a;2^- 10072^)2. 

5.  (y-50)2.  13.  (4.:^-xfy.  21.  (7a^^-2a;)2. 

6.  (y2  _  10)2.  14^  (^3^  _  ^^^y  22.  (11 0:2  _  13  ^y^ 

7.  (y  -  25z)2.      15.  (6a;y  -  5y2)2.         23.  (17a:yz  -  Syzdf. 

8.  (x^- 16  y2)2.    16.  (1-3  aby.  24.  (19  a;yz  -  2  a5c)2. 

Exercise  XXXVIII. 

1.  (a  +  b)(a-b).  5.  (a2  +  l)  («+ 1)  (a- 1). 

2.  (a  +  4)  (a  -  4).  6.  (a^+^*)  (^2+^2)  (ai-b)  (a-b), 

3.  (2  a  +  5)  (2  a  -  5).  7.  (a*+l)  (a'+l)  («+l)  (a-1). 

4.  (a2H-52)(a+5)(a-5).  8.  (6 ^  +  7y) (6 a; -  7y). 


14  ALGEBRA. 

9.  (10xi/+llab)(10xy-llab). 

10.  (1  +  7x)  (1  -  7x).  12.  (a-b  +  c)(a-b~  c). 

11.  {a^-\-bb){a'-bb).  13.  {x  +  a-b)(x-a  +  b). 

14.  (a  +  5-f  c  +  c?)(a+^  — c-c?). 

15.  ^xy.  17.  (2r  +  y  +  z)(a:-y-z). 

16.  {l  +  a-b)(l-a-\-b).  18.  (a:-y +  z)(a:-y- z). 

19.  (a +  25 -3c)  (a -25  + 3c). 

20.  {a~y-\-x-['z){a  —  y  —  x  —  z). 

21.  (2;  +  a;-y)(2;-^  +  y). 

22.  {x  —  y  -{-  z  -{-  d)  (x  —  y  —  z  —  d). 

23.  (a;  —  z  +  y  —  a)  (a;  —  z  —  y  +  a). 

24.  (a  +  5  +  c)(a  +  5  — c). 

25.  (a  —  5  +  ;r  —  y)  (a  —  5  —  a;  +  y). 

26.  (a:r+5y+l)(aa7  +  5y  — 1). 

27.  (l+^-y)(l-^+y).         30.  (:^  +  y+2)(a;-y). 

28.  12(a-l)(2a+l).  31.  {x-\-y)  {x-y-\-2). 

29.  (a-5  +  a;)(a-5-5;).     32.  (d  +  x -2y){d-x  +  2y). 

33.  (a  +  Z,  +  c)(a-5-c). 

34.  (2^  +  3a;-l)(2:r2-3a;+l). 

Exercise  XXXIX. 

1.  {a-b){a'+ab  +  b%  3.  (a;-7)(:^+7a;+49). 

2.  (a;-2)(a;2_|_2^^4)  4,  (y- 5)  (y^  +  Sy +  25). 

5.  (y-6)(y^  +  6y  +  36). 
*  6.  (2x-2>y){4:a?-{-^xy-\-^y'). 

7.  (4y-10z)(16y2  +  40yz  +  10022). 

8.  (9a;-8y)(81:^;2_^72:ry  +  64y2). 

9.  (3a-12)(9a2  +  36a+144). 

10.  (10a-115)(100a2  +  110a5  +  12152). 

Exercise  XL. 

1.  {x-\-y){o(^-xy-^lf).  3.  (^+6)(^-62:+36). 

2.  (^+2)(^-2a:+4).  4.  (y+4z)(y2-4yz+16z2). 


ANSWERS.  15 


5.  (U  +  bc)  (16  b'  -  20  be -j-  25c'). 

6.  (6a  +  8c)(36a2-48ac  +  64c2). 

7.  (9x  +  12i/)  (81a^-108xi/-i-14:4:y^). 

8.  (x  +  2/)  (;r'*  —  :r^3/  +  cc^y^  —  :?;?/^  -f  y"*). 

9.  (x  +  y)(ix'-x'y-i-xY-a^y'i-ar'y'-xy  +  y'). 
10.  (2^  +  3c)  (165*  -  24:b'c  +  36 Z^V-  546^3  +  81  c'). 

Exercise   XLI. 

1.  {a^  +  b')(a'~a'b'^  +  b'). 

2.  i^a"  +  b')  (a'  -  a«52  _|.  ^4^4  _  ^2^6  ^  ^s^ 

3.  (x^  +  y*)  (a:»  -  xY  +  y^). 

4.  (P  +  4c')(b'-Wc'-}-16c''). 

5.  (ri-2+l)(;r*-a:2_^^        7^  (4a2+:r2)(16a*-4aV+:^*) 

6.  (a'  +  1)  (a«  -  a*  +  1).      8.  (9  +  c')  (81  -  9  ^  +  c*). 

Exercise  XLII. 

1.  (a^  +  ab  +  b')  (a'  -  ab  -{-b'). 

2.  (Sa^  +  Sxy  +  2f)(Sc(^-Sxi/  +  2y'). 

3.  (Ax^  -\-Zxy-f)  (4:r2  _  3^,^  __  ^-^ 

4.  (9a2+7a^.+452)(9a2-7a5  +  452). 

5.  (9a2+10a5  +  4^>2)(9a2_10a5  +  4^>2). 

6.  (3a;2_|_2a:y+7y2)(3^-2:ry+7y^). 

7.  (5  a^  +  7a5  +  4  ^2)  (5  a^  -  7a5  +  4  b'). 

8.  (7m2  +  4 mw  +  9 n^)  (7m2  —  Amn-\-9n'). 

9.  (3a2-f  3ac+5^)(3a2-3ac4-5c2). 

10.  (7a2+13«^>  +  115')(7a2_13a6+115«). 

11.  (8a;2_^4^^_^9^2)^8:c2-4:ry  +  93/'). 

12.  (2:r2_|_5^^_3^)^2^-5^y-3y2). 

13.  (60^  +  xy  -  4:f)  (5x^  -  xy  -4^). 

14.  (9:z;2^4^y_^>)^9^^_4^^_^^^ 

Exercise  XLIII. 

1.  (4a;+l)(3rr--2).     3.  (4:r+l)(3:r-l).     5.  (a;+2)(3a:-2). 

2.  (3^-1) (4^-1).    4.  (a;4-l)(3:^-5}.       ^  (Sx-\-4:)(2x-l), 


16  ALGEBRA. 

7.  (4.x  +  1)  (^  +  3).  16.  (a  +  2x){a-  Sx). 

8.  (x  +  S){^x-l).  17.  (2a  +  55)(4a-35). 

9.  (2^+1)  (2a; -3).  18.  (3a- 2c)  (2a- 5c). 

10.  (a;-a)(a:-2a).  19.  (4a;  +  3y)  (2a;+ 7y). 

11.  (3a2  +  x")  (4a2  -  x").  20.  (4^;  +  83/)  (2x  -  7y). 

12.  (2a;  +  y)(a;  +  2y).  21.  (2.r  +  7^/)  (3a;-y). 
1.3.  (2aa:  +  l)(3aa;-l).  22.  (Ua~h)  (a- 2b). 

14.  (3 5  + a;)  (2 5 -3a:).  23.  (2c  -  d)  (c  -  Qd). 

15.  (2a; +1)  (2a; -I- 3).  24.  (23/  + 3z)  (3y-z).     • 

Exercise  XLIV. 

1.  (a  +  hf.  4.  (x  +  yy.  7.  (a;  +  y  +  z)2. 

2.  (a  +  iy.  5.  (a;-l)^  a  (x-y-zf. 
Z.(a-Vf.                  ^.(a-c)\  9.  (a  +  5~c)2. 

Exercise  XLV. 

1.  (2a;-y-3)(a;-2y-7).  8.  (2a;+y-3z)(3a;-y+3z). 

2.  (6a;— y+l)(a;— 6y— 1).  9.  (6a;— y-2;)(a;— y+62). 

3.  (3a:+2y+l)(2a;-3y-l).  10.  (5a:  — 3y  +  22)(a;  — y-z). 

4.  (5a;-3y+2)(a;-y+l).  11.  (ar+y+z)(2a;-3y-2z). 

5.  (a:+y-3)(2a;-3y-2).  12.  (2a:-3y+2z)(3a;-2y+3z). 

6.  (a:+5y— 3)(a;~5y— 7).  13.  (a;  —  y) (a;  —  y  +  5). 

7.  (2a;-y+z)(a;-2y-z).  14.  (2a;-y)(a;+3y-2z). 

Exercise  XLVI. 

1.  5(a;+l)(^-4).  9.  (a^  -\-  a-\-Y)(o?  -  a-\-V). 

2.  2a:«(a;-2)(a;-6).  10.  (x -y)(x^y~'z\ 

3.  3(a5  +  l)(«^-4).  11.  (a-h)(h~e). 

4.  (a  +  a:)(a  +  a;  +  4).  12.  (3 a; - y) (a; - z). 

5.  (a  —  h-\-c)(a  —  h  —  c).  13.  (a-\-x)(a  — x  —  h). 

6.  (x—y-\-c—d)(x-^—c-\-d).  14.  (a  —  x)(a  —  x-\-V). 

— r.  (2+a;+ar^)(2~a;~a;2).       15.  (a;-y)(3a:  +  3y-2). 
a  (a  +  5)(a-6-l).  16.  a:(a;2+l)(a;+l). 


ANSWERS.  17 


17.  (ax-l)(a^a^--ax-l).      18.  (:r+3y)(a:-3y)(3.r-2y). 
19.  (2x-l)(2a^  +  a^-]-l). 

21.  (x'-\-iJ')(x*-x'y'-{-y*). 

22.  ls  +  xXS-x)(9  +  Sx-i-x^)(9-3xr''^y 
83.  ^(x'  +  y^a^-xY  +  y'). 

24.  c(a2  +  c2)(a  +  c)(a-c).  27.  2a^(a;  +  y)(a7- Sy). 

25.  (a;  +  7)(:r-3).  28.  (2  a -5)1 

26.  3(a-2^))(a-55).  29.  4(2a;-5y)2. 

30.  :i:3y2^Q^_|_5^^)(6^_5j^) 

31.  (Sxf-bzy.  33.  (:r-y-z)2. 

32.  x(Ax'+l)(2x+l)(2x-l).      34.  (a-3^>)(a+26-4). 

35.  (a;  +  y  +  2)(a;  +  y-3). 

36.  (a^  +  2ab  +  b^  +  c^)  (a+  b  +  c)  (a  +  b  ~  c). 

37.  (a;  -  3y)  (a;  +  2y  -  4).       40.  (x  +  13)  (a;  +  7). 

38.  {l  +  x^)(l  —  x).  41.  (x  —  y)(x  —  z)(z  —  y). 

39.  (3:i:-2y)(:?;-32/).  42.   (:r  +  3)  (a:  -  8). 

43.  (x-{-y-z)(x  —  T/+z)(x  +  7/  +  z)(x  —  i/  —  z). 
44.    5a;(:iry+4z)(a:2/— 3z).       45.  (x^ -{-1)  (Sx  —  1). 
46.  (x  —  m  -}-  n)  (x  —  7n  —  n). 
417.  la  +  b  +  c)(a-\-b-c)(c+a-b)(c-a  +  b). 

48.  a^a'+l).  55.  :t-2(35+^)(25-3^). 

49.  (1-7A)2.  56.  (a -by. 

50.  (y  +  9)(y-13).  57.  (x  +  yf. 

51.  (:r  +  15)  (57  —  9).  58.  ab(a  —  b). 

52.  (2a-3H2c)(2a-35-2c).     59.  32/2(a;-3/)(3a;+2y). 

53.  la  +  ^c)(a-Sc+6b).  60.  (3a;  +  2y)(2a;4- 3y). 

54.  (3a:+2y-z)(3:r-2y+z).      61.  P(Sa+4:b)(2a-Sb). 

62.  (a  +  2^'-3c  +  c?)(a-2Z>  +  36'  +  c^). 

63.  (x'  +  4:f)(x-2y).  68.    (:^;  +  y  _  z)(:i;-y+2  +  l). 

64.  (2aa;-5)(2aa;-3^»).  69.  2(x +  7/)(x-{-y  +  a). 

65.  (3a:-2y)(6a;-4y+3).  70.  4:b(2a+Sc)(2a+c). 

66.  2(a:+3y)(a;— 2y+3z).  71.  (m  +  n)(w  -  w)  (^  -  ^). 

67.  (:r+y+l)(l-a;)(y-l).  72.  2a(3a;  +  y)(2:r- 3y). 


18  ALGEBRA. 


73.  2x(x+1){x  -  5).  75.  U(2x-y){^a^-{-2xy-\-fy 

74.  2xl2a-x){4:aj'-\-2ax-\-a^).  IQ.  x{l-^x){l  +  ?>x+^x'). 

77.  {^^J^f)(^x  +  y){x-y)(x'~^f-^y% 
{a?  —  xy  -\-  y^)  {x^  -\-xy-\-  y^). 

78.  llm  +  lln){lm-\ln).       80.  (2>z'+2)(4:z'-'S). 

79.  (4+92/2)(2+3y)(2-32/).    81.  {x^+l){x~l). 

82.  (x  +  y+l){x-y  +  l). 

83.  (7a  — 75  + 8m  — 8w) (7a -75 -8m +  8 w). 
g4    [{a-\-h  +  c){a-\-h-c)\  j{c  +  a-h){c-a  +  h)\ 

'  \  2ah  J    r  2ah  J  ' 

85.  (a: -8) (a; -45).  87.  {2h  -  e){a  +  h  -  c). 

86.  (:i:+2)(^-2)(a;-22/  +  l).         88.  5 rr  (5 ^  +  7y2)2. 

89.  a{a  +  h)  {a^  +  a5  +  5^)  {a^  -ab-\-  h^). 

90.  2;r  (a  +  c)  (a  -  c)  {p?  -ac-\-  c"). 

91.  (dx  +  2y-Sz)(2x-Sy~\-Sz). 

92.  (4a:-y+2;)(a;-2y-2).  94.  (a:- l)(a;3-a:2_3)^ 

93.  (Sa-b  +  2c)(a-2b  +  c).        95.  (5:?;-3y)(^-y-l). 

96.  (a  +  2h-^c-d)(a~2b-{-Sc-d). 

97.  (:r  +  2)  (a;  -  3)  (x  +  4:)(x-  5). 


EXEECISE   XL VII. 

1.  18a5c^. 

9.  7a; -4. 

17.  5y. 

2.  ITp^. 

10.  2a^xy(Sx-y). 

18.  a;(:r  +  l), 

3.  4a:2^V. 

11.  2abc. 

19.  3(a;-l), 

4.  30xy. 

12.  a;  -  3. 

20.  2(a-b). 

5.  a  —  5. 

13.  2  (a +  5). 

21.  ^  +  3/. 

6.  a— a;. 

14.  6^(a;-y). 

22.  x-y. 

7.  a  +  a;. 

15.  Sx(x  +  4:). 

23.  ^-1. 

8.  3a; +1. 

16.  c(a  — 5). 
Exercise  XLVIII. 

1.  5x-l. 

4.  (3a;+2)(3^-2). 

7. 

2:r-5. 

2.  a;  +  l. 

5.  35;(3:r2-^  +  l). 

8. 

a  (a  — 2a;). 

3.  a(3a-l). 

6.  2(a;-2). 

9. 

2a^-Sx^^ 

ANSWERS.  19 


10.  2a; -7.  15.  3 (a; +  3).  20.  ^r^-S^  +  l. 

11.  5x(6x-l).  16.  x^i-x7/+f.  21.   7:r+l. 

12.  y(3:r  — 4y).  17.  a -2.  22.  yi^s^-xy-'bf). 

13.  3rr-5.  18.  2(22/  +  5).  23.  2a<-^-4^4-a7-l. 

14.  4:c  +  l.  19.  3a;(2a;2_5)  24.  :?;-2a. 

Exercise  XLIX. 

1.  :rfl.  5.  y2_2y  +  5.  9.  2 (a; +  2/). 

2.  y  — 1.  6.  :r— 1.  10.  a?  —  xy -\- 'i/^ . 

4.  a; -2.  8.  a;  +  l. 

Exercise  L. 

1.  12  aV.  15.  (a7  +  y)2(a;-y)^. 

2.  naa^y\  16.  12(a;-y)2(rc3  +  3/3>) 

3.  a;2(a  +  ^).  17.  I20xy  {x  +  y)(x~y). 

4.  :r  (:r  +  1)  (x  -  1).  18.  (a;  +  2)  (a;  +  3)  (:r  +  4). 

5.  ala-{-h){a~h).  19.  (a- 3)  (a  +  4)  (a- 5). 

6.  (2a:  +  1)  (2^  -  1).  20.  {x  +  5)  {x  +  6)  (a:  +  7). 

7.  a^  +  h\  21.  (a;  +  2)  (a;  -  2)  {x  -  11). 

8.  (a;2+l)(a;+l)(a;-l).  22.  20o?h{a-h)(a-2b){a  +  U). 

9.  a;(r^+l)(^-l).  23.  240(a;+l)(a;-l)(a:+2)(a;-2). 

10.  xlx  +  V){3?-l).  24.  12a;2^(a;  +  y)2(a:-y)2. 

11.  (2«-l)(8a3  +  l).  25.  {a-h){h-c){c-a). 

12.  (a -5)  (a +  5)1  26.  (a-h){a-c){h-c). 

13.  4(l  +  a;)(l-aO.  27.  T'{x-l){x-2,){x-\-4:). 

14.  a;3_i  28.  12a:y(a;-y)^ 

29.  {a+h-\-c+d){a+h-c-d){a-{-c-h-d){a-^d-h-c). 

30.  60  (a; -3)  (a; -2)1 

Exercise  LI. 
1.  (2a;+l)(3a;-2)(7a:-l).    3.  (a;-12)(a:-3)(a:2__2)(^_j_3^_j_9) 
2.(a;  +  l)(a;-l)(a;  +  3)(2a;+l)(3a;-2). 


20  ALGEBRA. 


5.  2(3:r-22/)(3a;  +  5y)(4a;  +  32/). 

6.  2x(x'+l)(x-l)(x'-x-l). 

7.  12(2x  +  l)(2x-l)(Sx-{-2)(x-S)(x-A). 

8.  (x-l)(x-2)(x-S)(x-4:). 

10.  (x  +  y)(x-y)(x  +  2^)(x~2y).  12.  (1  ~  a)^ 

13.  (a  +  b  +  c)(a-  b  +  c^^ai-  b  -  c){b  +  c -  a). 

14.  c(3c2  +  y2)(4c4_3y)(^_2/). 

15.  (m  +  3)(m2-3m+l)(m^  +  l). 

16.  (5n2-l)(4w2  +  l)(5?z2  +  n+l). 

17.  (5  -  1)  (b'  -b^-  8)(4  6^-8  6+1).  18.  6  r  (r  +  1)'  (r  -  1)^ 


Exercise  LII. 
^^    a^  +  5a  +  5  ^i.    ^'  +  ^' 


4a;  a^  +  a  — 2  a 

2.    ^-=±.  12.    5^.  22.    -^. 

^+i  ,3    ^-2^  +  2  3(a  +  ^) 

*   a;-7  *         o?-2  '    ^ab(a-b) 

4.    ^.^-..  +  1.  14.         .^^-^^  „■     24.    ^L+A+f. 


4^+6aa;+9a2 


a 


5     ^!±^.  15  ^^±2^.  25  ^(^^-y) 

:r3-3/3  •  Sa  +  2b  '  y{Zx^2y) 

^    a^—a+1  ^g  a  —  b  —  c  ^  a  +  b  +  c-j-d 

a^-{-a-r-l  a-^b  —  c  a  —  b-\-c-\-d 

rj    a^  j^  ^  +  ^^-2  ^  Sx  +  2 

a-3  '  2a;2_^2a;+l  *  4^  +  5 

8.  ^Z:2  ^^  x_-d  ^^  ^-xy^f^ 

^  +  4  x-\-\  (^  —  y) 

9.  ^.nl.  19.  ^Zl^.  29.  :^2_|_^. 

10.    ^+^y-,<  20.       <^^~^)'    ■  30.  ^+^. 

a^—xy—'if'  0:^  —  3:^+1  a  —  b 


ANSWERS. 


21 


Exercise  LIII. 

1:^ 


1.  x  —  1,  6.  2a  — 3a; 


5  a  —  X 


2.  ?>x~lO-\ — ~  7.  12a;  +  3+      "'•^ 


a;  +  4  4^  —  1 

3.  3a;-6  +  ^^.  a2:^;  +  3-^     ^^ 


^  +  4  x  —  4: 

4.  a-2:.  +  -^-  9.  a+5+-2^. 

a4~^  ^  —  " 

5.  2a;+6  +  ^5-.  10.  ^^-l  +  ^^ili-^. 

^  —  3  5^  +  4^7  —  1 

Exercise  LIV. 

,      2y  ^    ax-\-2>  _    2a^-ah~¥ 

1.     ^— .  8.     -r— 15.     — 

x-\-y  2a  a  +  o 

2.  ^£L.  9.   -2^.  16.  3^  +  2^  +  V 

x-\-y  a  —  h  x-{-4: 

3.  ?!iii.  10.  n^s.  17.  ^+1. 

a;  a-{-b  x—\ 

.    2{a^-ax^a?)  „    a?-2xy-f  .^   x(3^-2x'-Z)_ 

a—x  x-\-y  x  —  2 

g    2(lla'^-20aZ)+8^^)     ^^    29a_  ^^    a^  +  2rg« 

5a-66  "       '     4    '  *    a  +  2a;' 

6.  ■^^.  13.    ^'  20.  ^+^y+y^. 

*  a-f  5  a  + 1  oj  +  a 


^^  2(19a-23a^-l)  ^^^ 


ar 


5-6a  a;-3 

Exercise  LV. 

9 a; -21     4rc-9  3   48a^-60ag     15a- IQg 

18     '        18     '  '         rna^c      '        60a*d? 

4a7-8.?/     3:r^-8a:y  ^    5  + 5a:     _6 


22  ALGEBRA. 


h~ 


{a  -—h)(a  —  c)  {h  —  c)     {a  —  b)  (a  —  c)(b-~  c) 
^x^{a  —  h)  xy 

7    30(4a;  +  l)     5(2:r-l)     S(Bx  +  2) 
'     15(a;-2)'    15(a;-2)'    15(a;-2y 
^    an  —  bmn     mnx     cm  —  bmn 


TTinx 


Exercise  LVI. 
^    163^+55a:+4a;.y-55y        ^    ll/-7a;^-4a?y-8a;^y' 
50a7  '  a^'i^ 

21x'-2x'y-l^xy~2^'if        ^    W(?-2ab^c-\-2a%c-o?d' 

12  a^  '         '  a%''(? 

180a^+54a^>-20a5^+331^>^    8.  6a^~3a^-2 

^    80a;«+64a;^+84a;+45  ^   a^^>  +  ^>^g  +  ag^ 

60^^                '  '            abc 

32^  +  9.y.  ^,^  ___!,. 

42  6y20 

Exercise  LVII. 
2a:-l  3        2  2a:-.v 

•  ar^-a;-30"  "  l-x"  *  (^-y/ 


:ir'-10^  +  21  \-\-x 

a^-o?b-ah^~b^ 


7. 


a^>  (a^  _  b"") 


Ix-VJ  g         2a;«  2a-35 

Exercise  LVIII. 
V  _2_.         "  2.  -i^.  3.  ?^±^.  4.  ^±^. 


1-a  l-a;*  1+r^  3/ 


AKSWEES.  ^2S 


'     :r3-95;2_|_26a;-24   '  *  (x  -  af 

7.  — ^  8.  0.         9.  2.        10.       ^    .      11    0. 

'  (a-a;)(a  +  2a;)  '         xy{x.  —  y) 

14  9a-105+^  16   ?-(^-^) 

17.  22;«-5^V+lQa;?/  +  5y^ 

(^  +  2/)'(^-y)' 

'  {a-\-h-\-c){a-\-h-c){a-\-c-h)  '    a'a^-by 

Exercise  LIX. 

1.  ^_.  3.  -1^.      •  5.  0. 

x  —  7/  ar  —  l 

2  _1_.  4  e+y        .     6  1 

'2  +  ^  •  3(l_f_2/)(l-_3^)         •  (^  +  a)(^  +  ^,) 

a«  +  2a'Z>-2a^>'  +  ^^  8    ^'~^^ 

9.  —5^ —  10.  0.  11.  0.  12.  0.  13.  0. 


2  +  x 


9a—10b  +  c  ,^  1 

14. -— — — -•  15 


(a  —  h)  (a  —  c)  (h  —  c)  '  z{z  —  x)  (y  —  z) 

Exercise  LX. 

1.  «^.            4.  i^.             7.  _M£.  10.  ^^. 

*  6c?              '    5Z>               '42;  *      a^ 

2.9a..         5.i2^.            a3=f/.  11.;^, 

9  ax                  ^jpf  {a  —  by 

3.  *•              a  —  a.r.          9. 12.  -^^ ^-^ ^ 

4^2'  ^ 


24  ALGEBRA. 


13.  ^1  la  -^^  23.  ^. 

14    (^+^)(^'+Q^^+^)     19  ^  +  ^         od    g-«  +  ^ 

15.  2a^(^-y).  20.  1.  25.  ^-^+^ 

^  x-\-a—  h 

to        oh  , 

^^-  ,q:4^2-  21.  i.  26.  1. 

17.  ■^^~^)^^~^).  22.  -^^.  27    "^-y-^ 

^  x-y  '  x-\-y-\-z 

EXEECISE   LXI. 

1.  1.  5    («-^)(^+^)^(a^+:r^)       o    l±f 

2.  ^i:i.  6.  V  10   :r4-l 

3.  _!-.  7.  ^±i.  11    1±^ 
a:  4-1                                2a;  '1  +  07 

,_    (.  +  «)(x-«)       3_       1     _  ^  j_ 

la  ^±^ 14.  (^+y)!. 

a^  —  2xy-\-'i^  2y 

^^    a{a'-\-2ah-\-2b')         ^^  1 

^  +  a  '  (a  +  ^)^  '  c{a—h  —  c) 

16.  a;.  18.  m  - 1.  20.  ^(^~^). 

7-4a: 


Exercise  LXII. 

1.  ^^^.         4.  ^ 7.  1  10  ^^-^. 

^  +  8  2(a;  +  lf  ■""•    a  +  5 

2.  1|.  5.  a;.  8.  -/^.        11.  ^. 

3.  46.  fi.  0.  8.  2. 


15.     «^ 


ANSWERS.  26 


12.  ^  +  3:r  +  3--  +  i.        13.  1.  14.  0.         15.  a. 

16.  i.  20.  a.  24.  '^'[tT^'^^ 

17.  -^ — ! ! — -•       21.  a{2a—x—y).  25.  , — -• 

oc-\-ac—ao  {x—z){y—z) 

4  1 

18.  m.  22.      .        _•  26. 


3(a;  +  l)  c(a-c)(6-c?) 

19.  (^L±k±Af..  23.  ^.  27.   -^S: 

2hc  2  (^  +  1)' 

Exercise  LXIII. 

1.  a;  =  16.  6.  ^  =  —  |.  11.  x  =  '^.  16.  a;  =  2. 

2.  a;  =  5.  7.  a;  =  9.  12.  a;  =  10.  17.  a:  =  7. 

3.  a;=i  8.  a;  =  li.  13.  rr  =  6.  18.  r^  =  9. 

4.  :r  =  1.  9.  X  =  7.  14.  X  =  12.  19.  ^  =  9. 

5.  37  =  8.  10.  :i:  =  — 6.  15.  37  =  5. 

Exercise  LXIV. 

1.37  =  8.  3.37  =  4.  5.    37=1^  7.37  =  2. 

2.    3r  =  9.  4.    37  =  20.  6.    37  =  4.  8.    37  =  3. 

9.    37  =-2.  10.    37=11. 

Exercise  LXV. 

1.    37  =  C.  6.    37  =  2.  11.    37  = !-—  • 

3  + 4a 

2,^^3c-2a  7.37  =  0.  12.3^  =  6-1. 

bh~c 

a  .r=  ,  '-{     ■    a  x=-^.       13.^=0*. 

4.  37=1.  9.  37=—  14.  x  =  bm. 

a-b  2a  +  b 


26 


ALGEBUA. 


16.  .=  24.  17.  ^=2rJ:. 

0  —  1  h 

la  37  =  --- -.  19.  x  =  - -.  20.  a;  =  — 

o  —  (f  bc-{-d  0 


Exercise  LXVI. 

1.  a;=7.  7.  a:  =  8.         13.  07=7.  19.  a7  =  3f. 

2.  a;  =  f  a  a;=-14.    14.  a;=a+-^.  20.  x       ""^ 


a-\-b       '         a-\-b 

3.  a?  =  -||.     9.07=14.       15.07  =  ^.  21.  o;  =  12. 

4.ar=li.       10.07  =  111.     16.  o:  =  0.  22.07  =  3. 

5.07  =  3.  11.07  =  2.  17.07=1.  23.  o:  =  -9. 

a  07  =  0.         12.  a;  =  —  2.     18.  07  =  1.  24.  07  =  |f. 

Exercise  LXVII. 

1.  24.      a  80.      5.  28,  32.        7.  76, 24.      9.  3456, 2304. 

2.  168.    4.  8.        6.  2f ,  47-f.      8.  960.         10.  36,  9. 

11.  420.        12.  85  gallons  wine,  35  gallons  water. 
13.  18  pounds  saltpeter,  3  pounds  sulphur,  3  pounds  char- 
coal.      14.  28,18.     15.  House,  $600;  garden,  $250. 
16.  17200.  18.  47,  23.  21.  19,  20,  21,  22. 

^^   a  +  b     a-b  19.  7, 32.  22.  42  years. 

2    '       2     '  20.  112, 96.  23.  20  years. 

24.  Father,  36  years ;  son,  12  years.  27.  2  days. 

25.  A,  28  years ;  B,  14  years.  28.  1-^  days. 

26.  Father,  40  years ;  son,  10  years.         29.  26|-  days. 

30.  A,  168  days  ;  B,  84  days;  0,  42  days.        31.  10|  days. 
32.  B  and  C,  21^  hours ;  A,  B,  and  C,  lO^f  hours. 

33.  10  days.  35.  A,  37|  minutes ;  B,  25  minutes. 

34.  40  minutes.  36.  4|-  hours.  37.  48  minutes. 
Oft            Q^^g                      39.  48f  minutes.        41.  7  miles. 

'  db  +  ac-{-bc  40.  42  hours. 


ANSWERS.  27 


40     ^g  44.  1st,  5  miles  per  hour ;  2d,  3  miles  per 

b-i-c  hour. 

43.  160  miles.        45.  175  miles. 

46.  I.  16^  minutes  past  three ;   II.  32^  minutes  past  six  ; 

III.  49-^  minutes  past  nine. 

47.  I.  32y\  minutes  past  three ;  II.  5y\  and  38y\  minutes 

past  four  ;  III.  21-^  and  54^^  minutes  past  7. 

48.  I.  38^^  minutes  past  one  ;   II.  54^^  minutes  past  four ; 

III.  10-^  minutes  past  eight. 

49.  5^^  minutes  past  two.  50.  700  leaps. 

51.  Greyhound,  450  leaps ;  hare,  600  leaps. 

52.  240  leaps.  53.  Length,  13  feet;  breadth,  8  feet. 
54.  56J  square  feet.        55.  Length,  27  in. ;  breadth,  14  in. 

56.  Length,  15  feet;  breadth,  11  feet. 

57.  Ill  pounds  tin ;  69  pounds  lead.  59.  189. 

58.  76  pounds  gold  ;  30  pounds  silver.  60.  6J  ounces. 
61.  330  feet.         62.  $228f .             63.  24  shots.  64.  60. 

65.  12,000  square  yards ;  56J  cents  per  square  yard. 

66.  20  seconds.     68.  103  sheep.  70.  10  gallons  from  1st  cask, 

67.  $55,500.         69.  $3700.  4  gallons  from  2d  cask. 

71.  60.  72.  1504.  73.  .39.  74.  9f  miles. 

75.  ^Zli.        76.  ^^^^  of  1st  kind;  ^^^  of  2d  kind. 
b-]-c  h~a  o—a 

Exercise  LXVIII. 

1.  x  =  2.  4.  a:  =  12.  7.  a;  =  29.         10.  a:  =  3. 
y  =  L              y  =  -3.  y  =  23.  y  =  2. 

2.  a;  =  2.  5.  rr  =  6.  3.  a;  =  8.  11.  a;  =  8. 
y  =  ~l.           2/  =  7.                y  =  4.  2/  =  20. 

3.  ^  =  2.  6.  a;  =  7.  9.  :r  =  7.  12.  a;  =  —  3. 
y  =  8.               y  =  9.                y=14.  3/  =  -2. 

13.  a;  =  —  2.  lA.  x  =  7. 


ALGEBRA. 


Exercise  LXIX. 

1.  a;  =10. 

4. 

a:  =  4. 

7.  07  =  20. 

10.  07  -=  2. 

y  =  7. 

y=l. 

y  =  10. 

y  =  2. 

2.  0^  =  17. 

5. 

37=5. 

8.  07  =  2. 

11.    07  =  2. 

2/ =19. 

y  =  5. 

y  =  -3. 

y  =  3. 

3.  ar  =  2. 

6. 

27=21. 

9.  07  =  8. 

12.  07  =  9. 

2^=13. 

y=12. 

y  =  4. 

y  =  ~3. 

13. 

^ 

=  - 

-5. 

14.    07  = 

--h 

y 

=  14. 

y  = 

=-i 

Exercise  LXX. 

1.  x  =  S. 

4. 

x  =  2. 

7.  07=12. 

10.  07  =  4. 

y  =  3. 

y=13. 

y  =  3. 

y  =  3. 

2.  ^  =  6. 

5. 

07  =  8. 

8.  07  =  2. 

11.    07=-:~2. 

y-3. 

y  =  l. 

y  =  e>. 

y  =  19. 

3.  ar  =  6. 

6. 

x  =  ^. 

9.  07  =  3. 

12.  07  =  7. 

y  =  8. 

y  =  4. 

y  =  5. 

y  =  -5. 

13. 

X 

=  2 

> 

14.    07 

=  11. 

y 

^  - 

-3. 

y-- 

-10. 

Exercise  LXXI. 

1.  x  =  l. 

7. 

07=11. 

13.  07=12. 

19.    07  =  4. 

y=7. 

2/  =  6. 

y  =  6. 

y  =  -l. 

2.  ^  =  2. 

8. 

07  =  3. 

14.  07  =  3. 

20.  07=^2. 

y  =  3. 

y  =  2. 

y  =  2. 

y  =  3. 

3.  a:  =  l. 

9. 

07=7. 

15.    07  =  9. 

21.  07  =  6. 

y  =  8. 

y  =  5. 

y  =  8. 

y  =  5. 

4.  x  =  9. 

10. 

07  =  — 5^. 

16.  07  =  5. 

22.  07=19L 

y  =  8. 

y  =  -9i- 

y  =  7. 

y  =  -17. 

5.  a;  =  -3. 

11. 

07=7. 

17.  07  =  7. 

23.  07  =  4. 

y  =  -2. 

y  =  8. 

y  =  2. 

y  =  3. 

6.  x=1. 

12. 

07  =  14. 

18.  07  =  9-^. 

24.  07=7. 

y  =  10. 

y  =  46. 
25.  07 

y 

y  =  2. 

=  21. 
=  20. 

y  =  9. 

ANSWERS.  29 


Exercise  LXXII. 

1.  ^=^.  a  x=\. 

2  aJ 

a-b  „       1 

2.  x  =  -^ — -■  3a 

«?-*?                                   3^-V^  +  d 
ar  —  cp  y  = — ' — • 

aq~op 

3.  a;  = 


4.    a; 


3/ 


hn-aq  ^^'    ^      ^_5' 

np  —  mq  ^ 

^  ap  —  hm  ~                ^  ~  a +  5' 

^^-^^'  11.   x^a-h. 

ce~hd  y  =  a-\-h. 


"^^■^^^  12.    x^         «^ 


^.  '  (a-l)(^-l) 


^—c  a— I 

y 


13.   X 


h.  X J- .  ,  7 


??^n'  +  '^^'w  y : 


2 
a  —  h 


2 


a/-6c^  14.    2: 


a 


c{ac-d)^  ^         2_52 

7    ^==_^^!^  15.   x^{a  +  hf. 
h^c-df  y^{a-h)\ 

J^hc_^  16.   x=a-\-c. 


30  ALGEBRA. 

,„  ah(?  19.    x=h  —  a. 

17.  x=a r-  . 

ac-\-b  y  =  a  —  o. 

y  ^  ah(a  +  h) 

V^^ 20.    x^= — ^^ — ' — ^• 

18.  x-=a-\-'b.  _     a?-h^ 
y  =  a-h,                                      ^      2(a2  +  62y 

Exercise  LXXIII. 
1  5.   x  =  ~\.  1 

2.  a;  =  :; —  V  =  — 

3.  a;  =  6.  a  +  ^ 


-^  c  ^      m 

a^  +  h^ 

9.    :r—  ' 


am-\-hn 
hm  —  an 

Exercise   LXXIV. 


1.    37  =  2. 

4.  :r  =  5. 

7.  a;  =  20. 

10.  37  =  4. 

2/  =  2. 

2  =  2. 

y  =  6. 

z  =  7. 

y  =  10. 

2  =  6. 

y  =  0. 

2  =  6. 

2.  x  =  b. 

5.  x==l. 

a  a:  =  |. 

11.    37  =  6. 

y  =  6. 

2  =  8. 

y  =  4. 

2  =  6. 

2  =  36i 

y  =  —  6. 

2  =  5. 

3.   a:=l. 

6.  x  =  2. 

9.  x==2. 

12.  37  =  46. 

y  =  2. 

2  =  3. 

2=10. 

y  =  3. 

2  =  1. 

y  =  -2I 

2  =  1. 

ANSWERS.  31 


13.  x=l. 
y  =  2. 
z  =  -3. 

15.  a:  =  — 1^. 

y  =  2i 
z  =  6i. 

17.  ^  =  i 

19.    37=6. 

y  =  -12. 

2=18. 

14.  x  =  l. 

y  =  -l. 

z  =  2. 

16.  :?;  =  -4. 
y  =  -3f. 
2  =  -4i 

18.  x  =  9. 
y=ll. 

2  =  13. 

20.  07  =  1. 

0.1      ^ 

2 

oo     ._^^+^-«^ 

a  +  6  25c 

..  y  = ! 

a  +  c  .                              2ac 

2  _a^+&"-g^ 


2a6 

25. 

x=lh 
y  =  -3i 

2=2,1,. 

27.  07  = 

2  = 

=  17. 
=  22. 
=  -25. 

_(a+5 

~c)(a+h) 

a- 

i-b-\-c 

b+c 
23.  x=i.  24.  37=  3J. 

y  =  i  y=if 

2=i.  ,2  =  1^. 

26.    3.  =  ^+^. 

2a 

_ah-\-ac-\-bc-h^-2^ 
^  2{¥-c') 

2>bc ~-  ab  —  ac  —  b^ 

^■~         20)^-0^) 

rto             '  /     I  z.\             (a—b—c)(a-{-b) 
28.  37  =  — (a+o).    v  =  ^^ -^ — ■ — ^• 


Exercise  LXXV. 

1.  41,  7.        2.  35,  11,  98.         3.  6,  4.         4.  4.78,  1.0095, 

5.  Father,  35  years;  son,  14  years. 

6.  Father,  40  years ;  son,  10  years. 

7.  A,  1 116;  B,  $166. 

8.  Wheat,  %1\  per  bushel;  barley,  $f  per  bushel. 
9.  A,  $31 ;  B,  $27.         10.  Horse,  $100  ;  cow,  $50. 

11.  -f.  14.  -^.  17.  28.  20.  75.  23.  717. 

12.  yV-  15.  I,  f .  18.  24.  21.  36.  24.  759. 

13.  |4.          16.  If.  19.  84.            22.  69.  25.  126. 
27.  7  i  hours,  4i  hours.  28.  3  miles  per  hour.  26.  432. 


32  ALGEBRA. 


29.  Distance,  20  miles ;  rate  per  hour,  8  miies. 

30.  Time  rowing  down,  4  hours  ;    time  rowing  up,  6  hours  ; 

rate  of  stream  per  hour,  -f  miles. 

31.  20  pounds  @  42  cents ;   10  pounds  @  54  cents. 

32.  40  pounds  @  90  cents ;  90  pounds  @  28  cents. 

33.  Eye,  20  bushels ;  wheat,  52  bushels. 

34.  A,  $3t;  B,  |3;  0,  $2i 

35.  A,  If  hours  ;  B,  3  J  hours ;  0,  7  hours. 

36.  54  boxes ;  36  bales.     37.  1st,  36  days  ;  2d,  45  days 

38.  21  crowns ;  63  guineas. 

39.  A,  20  days ;  B,  30  days  ;  0,  60  days. 

40.  A,  ^-^'        minutes;  ^^'  ^^^^^  ®  ^^- 


ac  +  hc-ab  $5000  @  4%. 

minutes;  42.  $20,000  @  5^ 

minutes.  43.  $25,000  @  6%. 


B,      ^    ^f^^ minutes;  42.  $20,000  @  5% 

ab  -^  DC  —  ac 

2abc 


ab-\-ac  —  be 

44.  Sum,  ""-^"^  dollars;  rate,  M%j}  <^,, 

n  —  m  an  —  bm 

45.  Sum,  —  dollars ;  rate, y~ — ^  % . 

a  —  b  ad  — be 

46.  1st,  $3000®  4%;  2d,  $4000®  5%;  3d,  $4500®  6%. 

47.  $6750  ®  3%  ;  $6000  @  5%. 

48.  3  per  cents  ®  $60 ;  4  per  cents  ®  $75. 

49.  3  per  cents  ®  $75 ;  3^  per  cents  ®  $88. 

50.  $2000  ®  2%  ;  $500  ®  4%.         51,  2660  square  yards. 

52.  Length,  14  feet ;  breadth,  10  feet. 

53.  Length,  90  yards ;  breadth,  60  yards. 

54.  A,  9  miles ;  B,  8-j^  miles.     55.  A,  12  miles ;  B,  10  miles. 

56.  Express-train,  45  miles ;  accommodation-train,  30  miles. 

57.  315  miles.  60.  Ox,$105;  sheep,  $8|;  lamb,  $5i. 

58.  30  miles.  61.  2  horses,  4  oxen,  94  sheep. 

59.  33i  miles  per  hour.  62.  A,  $50;  B,  $30;  C,$20. 
63.  A,$9f;  B,$5i;  C,$3.     64.  A,  $3i  •  B,$2A;C,$I. 


ANSWERS.  33 


Exercise  LXXVI 
3.  a;V-  4.  ^ 

6.  :^-^e>x'-{-12x  +  ^.         7.  :r*  -  80.-3 +  24a:2_  32-^  + 
&  a;^ +15a;^ 4- 90a;3  + 2700^^4- 405a: 4- 243. 


l.a«  2..-  3..y.  ..-':^.  5.?^^ 


a  a;«+15a;^  +  90a;3  + 2700^^  +  4050: +  243, 
9.  1 +  10o:  + 40^:2  _^80o:3  + 80  o:*  +  32o:«. 

10.  Bm^  -  12m^+67n  -  1.  12.  -  125aV3/«. 

11.  lea^SV^.  13.  49mVo:y. 

32o:V  15.  81a:*+108o.'3+54or^+12a:  +  l. 

*       243a«^>V'        16.  16a:*-32ao:8  +  24aV-8A  +  a*. 

17.  243  a^  +  810ax^  +  1080  aV  +  720  aV  +  240  a*a:  +  32  a*. 

18.  16o:*-32a:^y  +  24a;2^-8a:/  +  3/*. 

19.  o;V  -  12^/  +  60a;y  -  160o:y  +  240o;«y« 

-192o;yi  +  64a:«y^. 

20.  a^o'  -  21  a«6«  +  189  a'b''  -  945  a^b*  +  2835  a^J^ 

-  5103  a^b^  +  5103  aZ>  -  2187. 

21.  -243a^«^V.  22.  729a:y».  23.  -S125aWx''. 

27.  8-36a:+102o;2_i7x^_|.204o:*-144ar'  +  64a;«. 

28.  l-6a:+15a;2_20a;3^15^4_6^_|_^6 

29.  l-Sx  +  6a^-7c(^  +  6x*-Sa^  +  a^. 

80.  1  +  4a:  +  10:^2 +16a;3_f.  19^4  _|_  15^  ^10:r«  +  4a;^ +  0:8. 

Exercise  LXXVII. 

1.  ±a^  ±A  ±2a%,  4,  ax'f,  ±2a^bc',  -2a\ 

2.  -12c2^V    15^»V,  ±42c*0.        4.  2ai''^.  7.  170. 

•^    '        72«      "  32*        6.  92.  9.  1. 


Exercise  LXXVIII. 

1.  a^  +  2a-~l.  3.  2a^ -Sa^x- ax^. 

2.  oc^  —xy-\-if.  4.  3a,-^  —  4o:?/^  -  -  23/^. 


34  ALGEBRA. 


5.  2a*  +  4aV    -46'^  10.  2-3a-a2^2a^ 

6.  2a^-5x  +  S.  11.  Sa-b  +  6c-\-d. 

7.  4.ar'-2ah-\-2b^.  12.  6a^- ^a^i/ -ixf  +  y^. 

8.  3^-2ar^  +  Sx~4:.  13.  m*-2m3+3m*-4m+5. 

9.  a;3_2^y-f  2a:y2-3/^.  14.  :^-\xy-f. 

15.  a;2._2^y  +  y2-^.  17.  1+?  +  ^  +  ^. 


X 


x"  '  a;» 


16.  |-|a^  +  |.  18.  ^-1  +  ^.        19.  a;2  +  | 

o  z  0  a  2 


Exercise  LXXIX. 

1.  347 ;  69.4 ;  43.21 ;  37.89 ;  8.008. 

2.  129.63;  2.1319;  .4937;  .75416. 

3.  .9486 ;  2.4919 ;  .6557 ;  .0923 ;  4.1231 ; 

11.3578 ;  18.6348 

4.119.5633 ;  1.5811 ;  44.7213 ;  .5477 ; 

.1732 ;  10.5356 

5.  .0333 ;  .0632 ;  .0707 ;  1.4142 ;  2.2360 ; 

1.8027 ;  2.9325 

6.  i;  ^;  I;  H;  «;  i 

7.  .7071 ;    .8164 ;    .8660 ;    .1767 ;    .2338...,. ; 

.2190 ;    .9258 ;    .2886 

Exercise  LXXX. 

1.  aT  +  2y.  ^.  a?^x^\.  9.  \—x^x^—:i^. 

%  a -3.  6.  \-^x^^o?.  10.  o?-^Zah-^W. 

3.07  +  4.  7.  a2-2a-l.  \\.  (?-^hc-\-W. 

4.  ^-ax-o?.  8.  4ar'+4a;-l.  12.  2a^-\-^ab-ZW. 

Exercise  LXXXI. 

1.  65.  4.  9.6.  7.  11.7.  10.  192.  13.  32.1. 

2.  48.  5.  478.         8.  108.  11.  2.34.         14.  46.8. 

8.  64,         f,  U4.        9,  1,4,1,         1J3,  3,8i        J.5,  9.36, 


ANSWERS.  85 


16.  20.53.     17.  1.111.     18.  27.55.     19.  4.5045.  20.  1.7479. 

21.  1.3572 ;  .5848 ;  .2154 ;  1.6874 ;  .7368 

Exercise  LXXXII. 

1.  3o-5^>.         2.  l-x  +  ar^.         3.  2  —  x.  4.  3a;- 1. 

5.    1-7/. 


Exercise  LXXXIII. 


1.  ±7. 

4.  ±h 

7.   ±1. 

10.  ±2|. 

2.  ±3. 

5.  ±i. 

&  ±5. 

11.  ±5. 

3.   ±  V3. 

6.   ±  VIl 

'.            9.   ±3. 

12.  dz2. 

13.  ±2. 

1.^   -^ 

*^.-i- 

L4.    ±yj~: 

a 

ExEBOisE  LXXXIV. 

1.  2,  -  6. 

8.  1,-f 

15.  3,  0. 

22.  3, -f 

2.  8,  -  2. 

9.  14, -f 

16.  11,  0. 

23.  4,  -  1. 

3.  Hi  i- 

10.  h  i- 

17.  3,  - 1. 

24.  3,-f 

4.  8,-1. 

U.  6,  -  IJ. 

18.  lzfcV3. 

25.  2|,  0. 

5.  2i,  -1. 

12.  2,  |. 

19.  -3,-3|. 

26.  7,  —  2. 

6.  i-i- 

13.  ±  Via 

20.  5,  lif . 

27.  5,  2. 

7.  3,  -  2. 

14.  i.-4f. 

21.  10,  -  29. 

28.  3,  4. 

29, 

.  5,^4f 

30.  13,  ±  VS-^. 

Exercise  LXXXV. 

1.  —a±  a V2.               3   !^     _  T^^  40          w 

2.  2a±aVn.                '2'          2  "  '  ~2 
g^  a(a  +  h)     a{a-b)          ^    c±:^(^  +  ^ac  ^    ^ 

a  —  h'       a  +  h               '       2{a-\-h)  '  ac' 

3.  a   -.               9.  '^^^  —  <^~^c^  0.  10-  -  «>  —  ^• 

'  (t                  '     a  +  6  — 2c?   '  XL    l±V^ 


36  ALGEBRA. 


12.  a'-\-h^±(a-  h)  Va^  +  h\  14.  1,  -  a. 

^^-  2"^    '  "2"'  16.  ±V^. 

1,.  ,,  _ ^_M:^.  18.  4.,   - f .         19.  6d.3V?+4 

20.™,    -^.     2i.2!±i!±V(^:zi!Z+4^.      22.il. 
n         m^  2a6 

23.  ±m  Ji r.  35.  m  +  w,        ^ 


24. 


l_3a^2aVa  36.  2a-5,  a  +  2h. 

{a-Vf  ^    a±h     a-h 


a-\-h     a  —  h  a  —  b     a-\-h 

'a-h'    a  +  h  38.  2a  +  65,  a -85. 
26.  {in-\-7if,  —{m  —  nf. 

^   ^a-5b^    a-2b  39.  ^-^b,  -(5a+b). 


Gab  Sab 


4a2  +  95 


ac  be 

29.  a  +  |,    m-2a.  ^^-  "''  ^T^* 

3ft-25     2ft-35  42.  1, 

Q       '  5       '  a+b+c 

a  +  b  43.  1    H^szll. 


30. 


31.  2  a, 

2i 


c(a-—b) 
32.  a  —  b,  c.  44.-1,  _^~^^ 


33.  m±(n-p).  2a-2>b 

^  2    ^'  a-25         a  +  26 

ExEECisE  LXXXVI. 

1.  ±1.  ±2.  7.   1,-1,1.       _ 

2.  1,2,-3,4.  8.  i,  K-l=fcV-3Y 

3.  0, -i±V6.  9.  -^,  i(i  ±Vi::3)^ 

4.  0.5.-7.  10.    ±1,  i(dzl±V-3). 

5.  -2,  3,  -4,  5.  n.   0,  a,  ±5. 

6.  a,  ^  (— a~3:i:  V— ^c^-— oa-t-1).  12.  -1, 1  ±  -1 V  -iiw'-4». 


ANSWERS.                                               37 

Exercise  LXXXVII. 

1. 

^'-3:r  +  2  =  0.               5.    2a;^  +  lU  +  5  =  0. 

2. 

a;'-4a;-21  =  0.            6.    63a:^-32a:  -  63  =  0. 

3. 

6:^2- 5a; +  1-0.             7.    16a:* -153:i-'^  + 81  =  0. 

4. 

(Jx'  +  5x-6  =  0.            8.    a:(ar^~6a:^  +  lla;-6)  =  0. 

9.    x{4.x'-1x-S)  =  0. 

10.   a;'-4aar  +  3a2-4a5-4^^  =  0. 

11.  x''i-ax  —  6a''-\-5ab  —  h^  =  0. 

12.  a,**  — a'ar  — a:  — a'  +  a  =  0. 

13.  Rational  and  both  positive.  16.  Impossible. 

14.  Rational  and  of  opposite  signs.        17.  Surds  and  botli 

15.  Rational  and  of  opposite  signs.  negative. 

18.    Impossible.  19.  Rational  and  of  opposite  signs. 

20.    Rational,  equal  in  value,  and  both  negative. 


Exercise  LXXXVIII. 

1.  Max.,  13.         4  ^^^    (a -by        6.  Min.,  4. 

2.  Min.,  4a.           *  '       4               7.  Min.,  8. 

3.  Min.,  ±2.        5.  Max.,  =b^.                „    ,4-          4a5 

'             ,--  *                 8.   Mm., ; — 

9.    Max.,±Ki-  .    .                              ^"  +  *^ 

\a  11.    Each  10  inches.         12.  J. 

10.    Each  10  inches. 


Exercise  LXXXIX. 

1.  1,-2.  7.  ±V2j, 

2.  zfcl,  d=2.  8.  ±5^±2.  14.   VA±AV89. 

3.  ±3,  rbf  9.  -^-2,^^^.  15.  1,1,-3,5. 

4.  rbl,±i  10.  0,-J,-i  16.-1,2,-3,4. 

5.  1,  f  11.  Impossible.  17.   1-J-,  dr-J,  f 

6.  ±2,  ±i  12.  1,  V-4.  18.   i:l,  ±3. 


38 

ALGEBRA. 

Exercise  XO. 

1.  10,  11,  12. 

3.  1,  2, 

3.          5.  6. 

7.  $lli 

2.  3,4. 

4.  12. 

6.  12. 

8.  31i-  cents. 

9.  12  inches. 

14.  7.63 , 

12.36 

10.  121  yards, 

120  yards. 

15.  A,  36  hours ;  B,  45  hours. 

11.  16,  16,  28. 

16.  5  hours. 

7  hours. 

12.  li  yards. 

17.  3  hours. 

5  hours. 

13.  2.4142 

inches. 

18.  8  feet ;  4^  pounds. 

19.  39. 

21.  5  miles  per  hour. 

23.  $80. 

20.  15,  20. 

22.  4550. 

24.  $60, 

25.  100  shares®  $75. 

or  $40. 

26.  15,  9,  and  6  yards  ;  or, 

2|,  1  J,  and  f  yards. 

27.  5f  miles  per  hour. 

Exercise  XCI. 

1.  9,  4. 

6.  20,-6. 

11.  40,  9. 

16.4,-21. 

4,9. 

6,-20. 

9,40. 

1,-li. 

2.  25,  4. 

7.  10,  2. 

12.  6,  5. 

17.  3,  -  3|. 

4,25. 

2,10. 

5,6. 

4,-3if. 

3.  22,-3. 

8.  4,  2. 

13.  10,  2. 

18.  i. 

3,  -  22. 

2,4. 

2,10. 

*. 

4.  50,  -  5. 

9.  i,f 

14.  5,-3 

.         19.  i. 

5,  -  50. 

hh 

3,-5 

2. 

5.  13,  -  3. 

10.  3,  -  Iff 

15.  7,-4 

20.  ±  V2|. 

3.-13. 

-4,8ii, 

4,-7 

2=F  V2j. 

Exercise  XCII. 

1.  i  3,  =F  8. 

3.  ± 

4,  ±3V3. 

5.  ±  7,  zfc  5  Vi. 

±5. 

± 

5,  ±  V3. 

±  2,  =F  9  Vi 

2.  ±  2,  =b  V|.           4.  ± 

5Vf. 

6.   rt  2,  =fc  V2. 

±i  +2 

V|. 

v|. 

zt  4,  _-t  3  V2. 

ANSWERS.  39 


7.  ±  3,  ±  5VJ.  8.  ±  2,  ±  2f|.        9.  ±3.  la  zb  3. 

d=2,  ±V}.  ±3,  ±2^.            ±4.  ±1. 

Exercise  XCIII. 

1.  4,  2,  3  zt  V2r.  3.  4,  -  2,  0.                      6.  ±  3,  =b  2. 
2, 4,  3:=fV21,  2,-4,0.  i  2,  ±  3. 

2.  3,  2,  -3 ±  V3.  4.  4,  2,  ■|-(-13  ±  V377).  6.  3,  2. 
2,3,-3::fV3.  2,4,J(-13=fV377).  2,3. 


Exercise  XCIV. 

1.4,3.  ^3^^^  24.  ±3,  ±2. 

-3,-4.                      2  ±2,  d=:3. 

2.  ±5,  =h7Vi.                 a^j  25.  3,1. 
±  2,  ±  3V|.                2       ■  1,  3. 

3.  8,  -  3.  14.  3,  -  2.  26.  a,  0. 
If,  -  4.                         2,  -  3.  0,  -  a. 

4.  d=  7,  ±  1.  15.  i  4,  ±  lOf.  ^    ^^±1' 
±1,  ±7.                      ±9,  ±:2|.  *        a-5 

5.  2,-2^.  16.4,2.  ^  2ah 
-1, -2i|.                  2,4.  a-^> 

6.  8,  1.  17.  10,  -  10-J.  28.  ±  (a  +  h). 
-1,-8.                       15,-161  ^{a-h). 

7.  ±  f  la  3,  -  2.  29.  ±  4,  0. 
±f                              2,-3.  0,  ±4. 

a  2f ,  -  If.  19.  12,  10.  30.  ±  4,  ±  3. 

H,-2f.                       10,12.  ±3,  ±4. 

9.  ±  1.  20.  |(ajrV2^>^-a^).  31.  a +  5,  0. 

i-J.                               |(a+V252-a2).  a +  5,0. 

10.  4.  21.  ±lf,  =fciV3.  i{a-6±V(a+36X«-&)}. 
_  4.  ±  1|,  ifc^VS-  H«-^TV(a+36)(a-6)}. 

11.  i(±3=fc  V29).  22.  ±6.  32.  5,  f. 
j(4,3=fV29).            ±3.  3,-1  J. 

12.  ii,d.ivs^  ""'H-  ''•^^• 


40 


ALGEBRA. 


34. 


35.  3,  2,^(-3±- 
2,  3,  K-3h^- 

37.  6, -3f 
4, -3 A. 


38.  ± 


36.  5,-2,i(3±V=67). 
2,-5,K-3^V=67). 
/2a-3b\  I — 7 


fl^) 


i- 


Exercise  XOV. 


1.  7,5.  3.  i(3d=V5),  J(1±V5). 

2.  34,  43.  f  ±iV5,  ■|^^iV5. 
6.  15  miles;  3,  2^  and  4  miles.         7.   73,  37. 
9.  88  yards,  55  yards.        11.  8  feet,  10  feet. 

10.  290  yards.  12.  $3000,  7%  ;  $3500,  6% 


4.   16,  10 

R       2     3 

8.   16.  12. 


Exercise  XCVI 

1.  19,  8. 

10. 

8. 

20. 

91,  28. 

1,3. 

1. 

9,72. 

2.  1,4. 

11. 

21. 

21. 

8f ,  23-i. 

11,4. 

12. 

17i  70. 

3.  5. 

12. 

49.  . 

22. 

i 

4. 

22. 

1- 

4.  9,  19. 

13. 

32. 

23. 

6,  12,  18,  24. 

2,1. 

21. 

56,  42,  28,  14. 

5.  15,  7. 

14. 

19. 

8,  16,  24,  32. 

2,5. 

476. 

24. 

10. 

6.  4,  9. 

15. 

A,  A. 

A. 

1^- 

14. 

13,5. 

H.tV 

A, 

A- 

176. 

7.  6. 

16. 

8. 

25. 

149. 

2. 

17. 

59. 

26. 

6,  13. 

8.  28,  18,  8. 

18. 

3,  6. 

17,7. 

1,  8,  15. 

21,  2. 

27. 

14  half-crowns. 

9.  3. 

16,  32. 

9  4-penny-pieces. 

5. 

19. 

301. 

ANSWERS.  41 

29.  19  oxen.  30   34,  37,  40.  31.  905.  6Z.  34. 

1  sheep.  13,  8,  3.  3620.  17. 

80  hens.  1,  3,  5.  33.  14. 

Exercise  XOVII. 
8.  (a^-{-b^){(f+d^).  9.  m^  +  l.         '      10.  ia\a-b). 

Exercise  XOVIII. 

1.  x^;  x^ ;  x^  ;  a^  ;  a^ ;  a^ ;   a^b^. 

2.  x^y^z]  x^y^z^\  a^b^c;  5ab^c^ct^. 

3.  ^a^,  -VaFb-  ^V^' \  2>V^\ 
1  .  ^  .  6  y .  ^* .  ^xf 

d^'  x'lf'   a^  '  ]/'    ab^ 
6.  ^xyz"'^)  x'^y'^^z;  ab'^C^;  or^b'^<?\  x'^y^-,  x'^y"^, 

6.  (A  ;  b^  ;  ct ;  6^i  12.  a^  ;  ci ,  '^'^  ;  ai 

7.  m^  ;  Tit ;  c^ ;  y^  13.  a"^ ;  c'^  ;  m-^;  n^;  x, 

8.  a  ;  1  ;  y^  ;  a^i  14.  ^^  ;  q~^  ;  ar^y"* ;  a~l 

9.  atit.t;acU  ^5.  «  ;  ?;   JL ;  i, 

^J      :.»  ''-8'  9'82.^''     c' 

1*^  i:3'  ,T  ,^   27a'it.      04        aV  125 

37?  £"     y*  16. 


11    i--    -2^-L 


8        27a«6t    3^     64 


Exercise  XCIX. 

3.  .r- 3.^^  +  33:^-1. 

4.  lea  +  SaJbr+lOaH^i-lSaHr-SaH^ 

-4:aH7~baHr-9b. 

5.  1  4-  a^  5-2  _!_  ^4  j-4         6.  a*  5-"  --  4  -  4  a'^^^  -  a-'i*. 

7.  4:X'-''-~x'*-\-^x-^-^2x'^  +  x'^-{-l. 


42  ALGEBRA. 


8.  :r3n^^«2^«_|_^»^«_^^» 

9.  x^  —  x^y^~  x^  z^  -\-y^  —  y^  z^  +  z^-  W*  ^^  +  y^ 

13.  16a25-2;    a-2aU^  +  &;    a=^  +  2  +  a-2; 

4aZ>^-46  +  a-^it. 

14.  4  ;  10  ;  4  ;  i  ;  ^.  16.  Sa;-^  -  3^;-^^  +  y- 

15.  at-3a5^  +  3aU3-5.     17.  2x-^l-^x-^. 

18.  24.         20.  x^.  22.  a'"-\     24.   1. 

19.  a^K         21.  2aUi  +  llZ>.     23.  x^'i.        25.  a;(«-i)i'--y(i'-i)« 

Exercise  0. 

1.  V45;  Vl89;  VSOO ;  V^^;  -V^. 

2.  </8T^^;   \^32^;   </^^ ;  Wfab? ;  ^WWc, 

3.  Vfl;   V244|;  V^. 

4.  rry^V^;  2a-\/2ab-  2>ayV2^\  2V6;  ba'dVbd, 

5.  10^;  2a:/-V^20^;  3mV-v/4^;  7a*i*\/4^». 
a  (a-^.)^a;  5(a-5)V2. 

7.  4ac-{/5^5¥;,42VIl^;  27y\/3^ ;  55  V6. 

8.  iV5;  |V3;  |V2;  fV5^;   ^. 

9.  ^-^2^?;   i-v^;   25^8^6;  ^-sJUcxy. 
la  2|V5  ;  tI^  V21  ;  —^  ;  — - 

yt  G 

11.  a5a;t;  2a^h^^x;  Iba^h^y^. 

14.  7.071065  ;  42.42639  ;  0.707107  ;  0.1414213. 


ANSWERS.  48 


Exercise  CI. 

1.  3V7.  4.  24 V3;   V6.  7.  A\V30- 

2.  6V7;  5VlO;  9VJ.      5.  SJf  V7;  2.  8.  4|. 

3.  5^;  4^;  S-\/b.        6.  I-VS  ;   HV15.        9.  Y^^. 

Exercise  CII. 
1.  3V2;  1^;  2\/3.  7.  iv'225,000a^y^  4. 

'•^^"*^^*-  8.    «^,W. 

3.  3a/7  ;  4V2 ;  2-\/22.  ^V  ^  ' 

4.  3Vl9;  5^;  3^3.  9.  6  -  30 V3  +  24 VlO. 

5.  "lax-ypn:^^  \  a^fJ^\    10.  ^;  (2a5)* 

a"  '   0 

Exercise   GUI. 

1.  29V3;  30VI0+164V2.  6.  {a+l(dh)Vab. 

2.  13^/2  ;  33^/2.  7.  ahcVW?. 

3.  48  V2  ;  0.  8.  0. 

4.  |V3;  ftVT5.  9.  8a2V3^;  9^. 

11.  -^i  </8;  \V2-   -n-   </S. 

12.  -v^'2;   -</6;  V2;  V3;   ^;   V7. 

13.  a/2^;  -v/3^2.  aV2^;  2a« 

14.  2^2;  3^3;  4^2;  2v/2.         15.  a""* ;  a;-^ ;  pi;  a^. 


44  ALGEBRA. 


16.  aWa  +  baWb+lOabVa+lOabVb+bb'^V^i+bWb- 
m^  +  3  mxs/np?:!^  +  3  a^Vm^  +  x^Vx  ; 
a2V^-10rx2V^-f40a^>Va-80a^>V^+80//Va-32^/V/>. 

17.  64  a^-96  a» VS+60  a»-20  a^V^+3f  a«-f aVa4-^ ; 

_ a  64 

16 

64a;V''-192aJV*-v/^+240:rV"'^y^ 

-160a;V»  4- 60  a;*^/^  - 12  ^2/2^_^^^ 

^  2     ^    16 

19.  a^^-^d^tc^-t  -  3  a%"^  +  3  a^»-»c-i(^  t  -  c~^d^ ; 
a«"  -  4a*  +  6  -  4 a""  -\-  a'^" ; 


81 " 


20.  -f-r ^-f 


5tct     Z)2^^     3^3  ^5      27a«^>3' 
at       3a2    .       3  b^ 


^-^+¥«^^*-27i^;       21.  x^^  +  2>x'^y^  +  7f\ 


M     2^>H     4a^Z>*     Sa^ 


22.  l^2x-^-2>xr^-]-4:x-\ 


Exercise  CIV. 
1.  f(V7-V5);  |(2V5+V6);  5V2-6:  30  +  12V6. 

5  — <?         '            oj^—b           '           y(x~4:y) 
3.  1.154 ;  1.216 ;  3.576 ;  19.726 


ANSWERS.  45 


Exercise  OV. 

1.  19;  11.  4.  ~-2V5;  -2(3+18V=-T5). 

2.  6V^;  -axVby. 

3.  h-a-  - a'bWl.  y 


5  -•  _-x/irT-  — V^^ 


6.  2;   -V-3;  f 


Exercise  CVI. 


1.  3+V5.  7.  V6-V3.  13.  3VIT-2. 

2.  2V3+V5.  8.  7-3V5.  14.  3V5-2V3. 

3.  V7+V3.  9.  V10-V3.  15.  |VlO-l. 

4.  Vn+V5.  10.  3-V2.  16.  Va  +  6  +  Va-6. 

5.  V7-V2.  11.  2V3-V2.  17.  Va'-^'-^- 

6.  2(V3-V2).  12.  2V5-3V2.  18.  3V7-2-\/6. 

19.  {a  —  b)—2Vab. 


Exercise  CVII. 

1.  9. 

10.  rt  5,  ± : 

3V2. 

19.  2,  0. 

2.  4,  0. 

11.  i,~li 

20.  2,^,  |. 

3.  2  ±  iV6. 

12.  -28,0. 

21.  2  a. 

4.  6,25. 

5.  16, 1. 

13.  12,3. 

14.  1,  A. 

22.  -2a. 

23.  ±9V2. 

6.  5. 

15.  1. 

24.  4. 

7.  5,  ly^A- 

8.  «. 

16.  ±V65, 

17.  1\. 

d.V5. 

25.  z±r8. 

26.  2a/T8,  4. 

9.  ±24. 

18.  ±2a. 

^.  %f 

26.  4a^a^ 

29.  ±  3|, 

±U^-is. 

46 


ALGEBRA. 


Exercise  CVIII. 

1.  7,  ±4,-1.  6.  4,  1,  3|,  IJ. 

2.  f,-liyV(-13±V3l3)    7.  l,i(-3±V5). 

3.  2^,  -1,  U3±V=^.       8.  ±4,  d=V23. 

^   a  +  2^>     2Z>-9a  «•   U-l=tV5)' K-1±W53)» 

10.  6^^. 

12.   iff 


2a  2a 

5.  1, -1  J(li-^V4T).        11.  4,1. 

13.  i(l±V=^),  ±V=^. 

14.  5,-2^,  i(5±V97). 

15.  J,-|,|(1±V33). 


16. 


4a2+5* 


86^ 


17.  t 


2i0. 


18.  ±  2 a/2,  d=  i  V=^,  ±  i^(5  ±  V41)'. 

19.  -  8i,  5,  K4  =F  3  Vl09).   6f ,  -  2,  f  (-  3  ±  Vl09). 

20.  i(a  +  h  +  l)^iV^^^+l=PiVU  +  l. 

|(a  -  5)  =F  i  V4M^  ±  i  V4rfl. 

21.  ±2,  ±1,  |(±V6zhV2).     ±1,  ±2,  |(±V6^V2). 

22.  6,  2,  K-QiVSI).     2,  6,  |-(-9=fV33). 

23.  l-(a2  +  ^,2)  i  i-Vl0a2^2_  3^4_  3^4_ 
43  45 

1  (^2  ^  J2^)  ::p  i-VlOa^J^  -  3  a*- 3  ^>^ 
4b  46 

24.  3,  2,  J(5d=V6l),  -liVlO,  _1±V^. 

-2,  -3,  i(-5±V6l),  IztVlO,  1±V^. 

25.  3  zb  2V2,  0.     (3  ±  2 V2)2,  0. 

26.  6,  -  4i    12,  -  9.  27.  ±  16,  ±4.    rh  4.  =fc  16 


9a 
^.  ±2 


28.  ?^(^a±-Va^  +  a),  ^(a±Va2  +  9a). 
V  30.  9,  0. 


T\.r*  O  TY  JliXVO. 

1 1 

Exercise  CIX. 

1.  0.7781. 

12.  2.6232. 

23. 

0.0970. 

2.  1.1761. 

13.  1.0791. 

24. 

1.5741. 

3.  1.3222. 

14.  1.7781. 

25. 

0.3222. 

4.  1.1461. 

15.  1.8751. 

26. 

1.2040. 

5.  1.5441. 

16.  0.8751. 

27. 

8.7481  - 10. 

6.  0.9542. 

17.  8.3222  - 10. 

28. 

9.7993  - 10. 

7.  0.9030. 

18.  9.5441-10. 

29. 

0.2431. 

a  1.6902. 

19.  7.5441-10. 

30. 

2.0212. 

9.  1.3980. 

20.  7.6020-10. 

31. 

8.0212-10. 

10.  1.4771. 

21.  8.6990-10. 

32. 

0.0212. 

11.  1.6232. 

22.  1.0970. 
Exercise  CX. 

1.  0.9030. 

7. 

0.1003.          13.  0.2258. 

19.  2.9579. 

2.  1.3980. 

8. 

0.3495.          14.  0.4660. 

20.  0.6361. 

3.  3.3804. 

9. 

0.0596.          15.  0.2045. 

21.  2.4465. 

4.  3.8168. 

10. 

0.1690.          16.  0.2415. 

22.  0.4730. 

5.  2.5353. 

11. 

0.1398.          17.  1.1650. 

23.  0.5243. 

6.  3.4950. 

12. 

0.0768.          18.  0.3904. 

24.  1.3280. 

25.  1.1569. 

Exercise  CXI. 

1.  9.6020- 

-10. 

11.  0.1761. 

21 

.  0.4559. 

2.  9.4559- 

-10. 

12.  1.1461. 

22 

.  8.8539-10. 

3.  9.7781  - 

-10. 

13.  8.2219-10. 

23 

.  7.6320-10. 

4.  9.6320- 

-10. 

14.  7.3980-10. 

24 

.  7.5441-10. 

5.  9.8539- 

-10. 

15.  8.1461  - 10. 

25 

.  5.1707-10. 

6.  0.1461. 

16.  1.8539. 

26 

.  4.8293. 

7.  0.2219. 

17.  2.6320. 

27 

.  5.9333. 

8.  0.3980. 

18.  6.6320-10. 

28 

.  4.1040. 

9.  0.3680. 

19.  1.2219. 

29 

,  2.8627  -   10. 

10,  0,5441. 

^20.  9.544X  - 10, 

30 

.  7.1375. 

iS 


ALGEBRA. 


1.  1.7782. 

2.  2.0043. 

3.  2.9996. 

4.  3.9957. 

5.  3.7329. 

6.  3.5775. 

7.  4.7350. 
a  4.9581. 
9.  4.0000. 

10.  4.0004. 

11.  4.8490. 

12.  4.0802. 

13.  7.9943- 

14.  9.9430-10. 

15.  0.0000. 

16.  2.9430. 

17.  1.8686. 

18.  0.8494. 

19.  8.9528  - 10. 

20.  1.9990. 

21.  17,730. 

22.  4295. 

23.  339.6. 

24.  88.8. 

25.  53,910. 

26.  93.98. 

27.  0.7586. 
28    1.9. 

28.  0.1099. 


Exercise  CXII. 

30.  0.695. 

31.  0.001051. 

32.  0.0455. 

33.  41.64. 

34.  0.02989. 

35.  0.3131. 

36.  8.294. 


60. 
61. 
62. 
63. 
64. 
65. 
66. 
67. 


37.  -28.47. 

38.  -0.3144. 

39.  0.0104.  68. 

40.  -660,600.  69. 

41.  1.308.  70. 
10.    42.  0.6307.  71. 

43.  105.4.  72. 

44.  0.7167.  7a 

45.  -111.3.  74. 

46.  0.001086.  75. 

47.  0.7277.  76. 

48.  0.2084.  77. 

49.  6.963.  78. 

50.  -5.219.  79. 

51.  221.5.  80. 

52.  1.416.  81. 

53.  2.257.  82. 

54.  15.81.  83. 

55.  0.2387.  84. 

56.  0.03593.  85. 

57.  7.212.  86. 

58.  0.0001174. 
87.  0.000000000004936. 


12940. 

99.82. 

2.381. 

2.155. 

282.6. 

1.706. 

146.8. 

1.938. 

1.913. 

1.616. 

9.218. 

2.468. 

5.487. 

1.523. 

0.2356. 

1.293. 

0.7397. 

0.02558. 

1.568. 

6.955. 

0.002431. 

0.3525. 

0.0005218 

0.01804. 

33.04. 

28.39. 

12.58. 

0.5096. 


ANSWERS.  49 


Exercise  CXIII. 


1.  24:35.       4.  15:16; 

23 

:25 

;  10:11;  3:4.       6.  4 

:15, 

2.  10:9.         5.  a  +  b-.a 

~ 

b>. 

a'  +  b':a'-P.      7.  2 

:3. 

8.  a*  +  a^x^  +  x*:a*-x*. 

10.  1  :  1.               13.  6, 

8. 

9.  cc'-llx  +  28:a^. 

11.  18,  27.           14.  4 

:1. 

15.  1  :  5. 

17  '^p-'^q 

16.  26,  40,  A  to  B. 

q-p 

52,  60,  B  to  0. 

18.  1240,  680  rods. 

Exercise  CXIV. 


16.   3,  f 

lo    b(a+by 

19.  ±9.    ±3. 

17.  a-\-b,  i{a~b). 

(a --by 

20.   1.    4. 

21.  dz2. 

25.  440  and  362  yards  permin. 

22.  8:7. 

26.  45  and  30  miles  per  hour. 

23.  A's, 

$200;B's,  $150. 

27.  70  feet. 

24.  6. 

28.  3:7;  3/^ 

:7A. 

Exercise  CXV. 

1.  9|. 

6.  64X'=9Y'. 

10.  Vol.,  1611; 

14.  87.1 dys. 

2.  f 

7.  X^=i^. 

dis.,  402i. 

15.  1  inch. 

3.   15. 

ys 

11.  9i. 

16.   1  :  1. 

4.   5. 

8.   5. 

12.  20^  feet. 

17.  10. 

5.  S^. 

9.  8. 

13.  516|oz. 

18.  18  feet. 

Exercise  OXVI. 

1.  53. 

13.    -29.    a  +  21b.    -1.    0. 

2.  4^. 

3.  7i;  6;a^  4.  5f  10,  14^;  f  4i,  Sf  12f 


60  ALGEBRA. 


^'  ^^^  7.  185.    35.    ^(3n-l).   -28.    3361. 

6.-7.  2 

8.  1,  2f,  3|,  5^,  6f ,  8.       9.  9.      10.  -  10.      11.  6,  14,  22. 

12.  6.  15.  4,  15,  26, 18.  5,  7,  9. 

13.  12.  16.  3,  5,  7;  or  7,  5,  3.  19.  -  f,  I,  IJ. 

14.  5.  17.  8.  20.  14. 

21.  10  days.      22.  234.  23.  2,  6,  10,  14;  or  14,  10,  6,  2. 

25.  7  days.  27.  3,6, 15;  or  15,  12, 3. 

36.  6433i  feet.  28.  10  seconds;  HHe  distance. 

Exercise  CXVII. 

1.  1458.    96.    yfg.    -128.    x^\    201  a' m\ 

2.  Qx'fVlbi.       3.  4.     '4.  20,  50;  28,  56,  112.      5.  5th. 

6.  3  ;  192.  7.  765.    547.    15M.  1953.1.  ?5^. 

^  256 

8.  10%.       9.3.        10,  llfl.        12.  i  1,2, 4.        13.  ^V 

14.  12,  18,  27.  16.  0.59049.  18.  1,  3,  9. 

15.  1,  2i.  17.  2,  4,  6,  9.  19.  49,  1. 
20.  1,  3,  9,  27;   or  27,  9,  3,  1.  21.  139. 

22.  8.     H.     i.     ^.    ^\.     H.     i     If.     il    ||. 

Exercise   CXVIIL' 

1.  2|,  3,  4,  6.  4.  13-^,  Uj\ 17|,  183%.     7.  8,  2. 

2.  9,1;  or  f ,  -7^.    5.  6th.  9.   104,  234* 

3.  12.  6.  6,  6|,  6f ,  7y%.  10.  2,  3,  6. 

For  answers  to  Exercises  CXIX.  and  CXX.  of  Elements  of  Alge- 
bra, see  answers  to  CXX.  and  CXXI.  of  this  book  ;  and  for  answers 
to  Exercise  CXXI.  pf  Elements  of  Algebra,  see  Appendix- 


ANSWERS.  61 


Exercise  CXIX. 


1.  454.053,600. 

8.  1956. 

15.  33. 

2.  6. 

9.  109,600 

16.  720. 

3.  6. 

10.  2520. 

17.  400,400. 

4.  12. 

11.  420. 

18.  6160. 

5.  4815;  969;  3876. 

12.  5040. 

19.  161,700. 

6.  95,040. 

13.  244. 

20.  1,368,000. 

7.  5040;  144. 

14.  6720. 

21.  15. 

\mx\n 

23. 
24. 

1814400.25.  5775, 

(|rX|m  — r)(|sX 

\n-s) 

11;  3.      26.  36. 

^       \^      .      m 

28.  80. 

29.  |9  |2. 

30. 
31. 

325.         32.  7. 

'•  [20  |70'  [19  |70' 

645.         33.  52. 

Exercise  CXX. 

1.  1 +  10a;  + 40:^2 +  80a;3_|.  80 r?;*  +  32a^. 

2.  o(^-  24:r^  +  252:r«  -  1512:^;^  +  5670a;*  -  13,608a;3 

+  20,412a;2  _  17496a;  -f  6561. 

3.  l^x^-ma^y-\-2lQxhf-2mxif-{-^ly\ 


4.  S-\2x-\-Qa?-a?. 

.       15  y      45  y^      135?/     405  y*     243  .y^ 
4  8  32  256        1024' 

6.  l_3.  +  4.--— +  —  -  —  +  — 

7.  -  14,080.000  rry.  ""  243  "^  729  ~  19683' 


52 


ALGEBRA. 


g     85a:y  11.    70. 

^^^^  12.    -252. 

S5x     ,  35y 
10.    -115,200 a; V-  7"  '       "7~* 

1X2X (r-1)       ^^"^^         "^    • 

1X2X (r-1)      '^     ^ 

16.  H^-^) (^-^-^)a----3af-^3 

1X2X (r  +  3) 

17.  -±=a":r''.  18.    0. 
(^7 


Exercise  CXXI. 

^2      8^16 

2.  l  +  lor-iar'  +  A^"- 

,        !  f  1   ,  3:7:       3a;'    ,     5a;' 

3.  a^\  1+  + 


4a      32a''      128r/  j 

4.  l  +  4x  +  10a;'  +  20:c'  + 

If  n  be  an  integer,  in  the  expansions  of  (l  +  a:)*  or  (1  — a?)-**,  the 
coefficients  are  all  positive;  of  (1  — .r)"  or  (1  +  x)-^  they  are  alter- 
nately positive  and  negative. 

5      5aV  ,  15  ax*      5x^   , 

5.  ,5___  +  _^^.____  + 

6.  X-'  -  f  :r-V  +  \^^~'y'  -  H^"^?/'  + 

7.  (2a;)-i;i  +  |:r-V  +  AV^''2/'+TVA^~'y'  + 1- 

8.  l-x-2x''~e>2^- 


ANSWERS.  53 

*  8a4     "^Sa      128^2      1024 a^      )' 

10.  l  +  |y  +  ^Y  +  .%5^  + 

11.  l  +  |.r  +  f:r2-it-r«  + 

12.  l-|a;  +  ¥a;2-H^  + 

^       _        1X3X5       (2.-5)  ^3^^., 

h'-i  X  2-^ 

The  factor  (—  l)*"  is  introduced  because  the  rth  term  will  be  positive 
or  negative  according  as  r  is  even  or  odd, 

1x2 
16.  8.06224.  16.  1.01098.  17.  2.002224. 

la  l  +  ^x-^x^-\^x^+ 19.  351. 

Exercise  CXXII. 

1.   h,  i  I.  2.  f.  3.  -iJ^.  4.  f  f 

5.  7  to  2  against  the  event.  6.  -f . 

7.  h  10.  I".         14.  Their  chances  are  i,  i,  ^. 

8.  ^-^.     12.  ^\.       15.  $40.         17.  34  to  43. 

9.  f  13.  I  16.  2  to  3.      18.  ^^,  ^-  i^,  ^,  ff. 
19.   ^At-       20.  t\,  3-b.  23.  3-^.  24.  ^2^,  -1%.  fi. 

Exercise  CXXIII. 

1.  f.                        7.  rl^.  15.  f. 

2.  y-.                         8.  49  to  15  against.  16.  ff . 
n                           9.  ylfriy.  17.  T^. 

4.  1  -  (tt)'.  11.  Tfir.  t'A.  j%- 


54  ALGEBRA. 


6.  5  to  3  against.    12.  ^,  ■^,  j\.  19.  {{ 

7  to  1  against.    13.  ^,  ^,  fff,  |||.  20.  1  -  (ff )^ 

6.  6  to  1.                 14.  7  to  2.  21.  y^. 

250    276     7750            24.  ^.  26.  131  to  112. 

A5  '        A5  '  f^5     ■ 

25    1-15M  07   3497x5^ 

23.  ^,  ^,  ^,  -i^.                    '            5«   ■  ^''         8«       • 

28.   1||.         29.  As  16  :  12  :  9.              30.  $0.37i     7  to  1. 

31.  He  may  expect  a  loss  of  $0.25. 

32.  $6.93,  $12.86,  $3.96.                  '  ' 

33.  +$2e,  -lit.  -$2i|;  +$2|,  +$f  -$2A. 

Exercise  CXXIV. 
1.  Between  48  and  49.  12.  4|%,  nearly. 


(ii\  log ^- log ^ 
^              logi^ 

13.  $24.50. 

14.  4%. 

15.  $7298. 

3.  $.00356. 

16.  A  little  more  than  9  yrs. 

4.  ^  X  4.310. 

17.  A  little  less  than  17  yrs. 

5.  $445.20. 

18.  $968.45. 

6.  $20,255,  nearly. 

19.  51.37. 

7.  5.358. 

20.  4.708. 

8.  $34.30. 

21.  ^%. 

9.  22. 

22.  2,428,900. 

10.  115,  nearly. 

23.  $27,657. 

11.  5%,  nearly. 

- 

ANSWERS.  55 


Exercise  CXXV. 
1 


3  + 


3  +  -1 


2-f 


1+       ' 


1  +  -1  •       '^'      ' 


1+  ^ 


2  +  i.   3  +  i    i.   9+i    1    1    i. 

The  fifth  convergents  are  .1| ;  j^  ;  f  || ;  ^J^  ;  ^i- 
1  2+       1 


5  +  —^  4-+      ^ 


2  +  ^  6  +  1 


1 


2  +  -1 


4+-1  ^'        1 


The  third  convergents  are  ^  ;  f|-|- ;  ff  ;  |f. 
3.     4  + ^  4  +  —  ^ 


1  +  -^ 


1  +  — ^  -  •         1 


2  +  — J—  4+       ^ 


l  +  -i-  ■         2-f      ^ 


8  4- etc.  8  + etc. 


56  ALGEBRA. 


l  +  -i-y  2  +  -1- 

10  + etc.  ^14  + etc. 

5.  H-  8.  ^. 

6.  fW  9.  ¥;  M;  Hi- 

10.  15  and  16,  61  and  65,  76  and  81,  etc. 

11.  Vl5;  i(Vl97-13);  ^(Vl093-13). 

12.  Limit  of  error  =  .00012.  13.  {,  ^,  ^,  ■^, .. 

14.|,f,if  Jfi 


Exercise  CXXVI. 
2"*"  4  "^   8  "^  16  "^ *  2     4"^   8        16'"^ 


•  i"^16"^64       256      

4.  l-^-a;8+:?;«+ 7.  4:^+2:^2- 8^-22^*- ■ 

5.  \-\-2x-\-a?~^a?— 8.  y-'if  +  f-y*  + 

6.  5-17a:+56:r2-185a;3_|_ 9^  y^2y^+bi/-\-14:y^-\-"' 


•^3       15       315^ 

^      2^3      4^ 
l  +  2:r  3-7a; 


12. 


l-^+a;^'    i._3^^^2'    4  +  r-5a,-^ 


ANSWERS.  57 


,3    _3_4-_l-  15    _-§ L_ 

•  ^_|_4"^^_5  •  3(a;-5)      3(2:^7-1) 

14. -^_    6  la       4       ■       ^ 


a:  +  3     :i;  +  4  7(^7+2)      7(a7-5) 

17.  -J___^+A_. 
*  a;  —  1      a;^  +  a;4- 1 

18     3-1-,^+        3 


4a;      8(a:-2)      8(a:  +  2) 

4  4  71 

19.  ~ —^+         '^ 


25a;      ba^      25(a;  +  5) 
^.11      +      2^^4       10 


3(a;  +  l)      (x-iy      S(x  +  2) 

21         10  4a;  +  7 

*  3(a;+l)      3(a;2-a;+l) 

Exercise  CXXVII. 

1.  77323.  3.    631.  5.   2.425.  7.    68. 

2.  11860.  4.    120.  6.    766.5.  8.    255. 

9.    3n(n  +  l)(n  +  3).       10.    ■J-n(n+l)(4n-l). 

Exercise  CXXVIII. 


1.   11.              4.   460.                  7.   4095. 

10.   5525. 

2.   11879.        5.    1540;  2870.     8.    1840. 

11.   4970. 

3.   5610.          6.    190.                   9.   25. 

Exercise  CXXIX. 

^1/111        1           1           1    Y 
*   3V1      2^3      71  +  1      n  +  2      n  +  3y' 

11 

18* 

.    U                 1                                      3 

1 

2(n  +  2)(n+3)      4(n+l)(n-f2)(7i+3)(n-H)  '  96 


58  ALGEBRA. 


3    lA              1          \.   I 

'   8^4      2(n  +  l){n+2)J'  32 

^   5             3w  +  5        .  5 

6      (n  +  2)(n  +  3)'  6 

5     1                   1               .1 

'   4      2(7i  +  l)(7i  +  2)'  4 

Exercise  CXXX. 

1. 

2r*-f  3r8+8r2+7r  +  6.              3.    12. 

5. 

1111. 

2. 

2rH7r2  +  20r2  4-23r  +  20.        4.    9. 

7.   20.2444.                  8.   2709316. 

Exercise  OXXXI. 

6. 

3. 

1. 

x=7,  y  =  -2.                    ^2.(^  =  1^'  y 
x  =  4c,  y  =  2.                              \x  =  2,  y  = 

=  2. 

2. 

=  10. 

3. 
4. 
5. 
6. 

7. 
8. 
9. 

x=l,  y^-1.                    13  1^=13,  2/ 
:.  =  ^,   y=5.                            l^  =  -3,  , 
^  =  8,  y  =  4.                          14.|^  =  ^'2/^ 

^           -,                                U  =  -3, 
a:  =  2,  y=l. 

"...=-.       "■{::!:;;: 

^  =  29,  y  =  23.                            r^  =  -12 
.r  =  2,  y  =  -l.                       ^^•U  =  -3,' 

=  3. 

-f- 

-4. 
-9. 

y  = 
y^ 

-13. 

-4. 

14. 
-4. 

10. 

x  =  ~l,  y  =  -\.                    17.  .r  =  2,    y  = 

^¥- 

11. 

x==\,  y  =  \.                           18.  r.  =  6,  y- 

:9. 

Exercise  CXXXIV. 

1.  a.'»-.^•2 -44:2; +  84  =  0. 

2.  x^ ~ CI? -Vix" ■\- x-\-\2^0. 

3.  .7;^  +  6a7^- 13:^3 -78a;2_|_ 35^  _l_ 216  =  0. 

4.  .^^  +  0.075  :i-2-0. 105  .^•  + 0.01=0. 
6.  .i-«-ll:r2  +  40^-50  =  0. 


ANSWEES. 


59 


Exercise  CXXXV. 
1.  x^-Sx'-10x-\-24:  =  0.      4.  :?;3_i8a;2^108a;-216=-0. 

3.  a;^-4:x  =  0. 


5.  x'-5x'  +  4:  =  0. 

6.  4a;*-17ar^  +  4=0. 


.  Exercise  OXXXVL 

1.  -3,  -8.  6.  i(-3±Vl7),  1,  f 

2.  -6,-17.  7.  f5,  ^b. 

3.  ±a,  ±aV3.  8.  ±2,  1±V^,  — 1± 

4.  -2,  i±v^,  =bV^.  9.  ^,  ±v=n:. 

5.  I,  -i.  10.  I,  |(-lq=V=^),  0. 


Exercise  CXXXVII. 

1.  4,K-7±V37).  6.  -4,i(7±^ 

2.  -2,  3±V5.  7.  2,-3,-8. 

3.  3,  2±V^.  a  2,  ±5. 

4.  3,  3,  2.  9.  1,  9,  -7. 

5.  -5,  i-(5zfcV29).  10.  -3,  7,  -4. 

Exercise  CXXXVIII. 


T53). 


1.  1,  2,  3,  -4. 

2.  2,-1,  2±2V2. 

3.  2,  4, -1±V^. 

4.  3±V2,  3±V'2. 

5.  1,  1,  2,  -4. 


1. 


^=3. 


6.  4,  -2,  -1± 

7.  2,  3,  IrbV^. 

8.  2,-2,  -l=bV6. 

9.  13,-1,-1,-7. 

10.  2±:V=nLl,  ^(7=bV37). 


Exercise  CXL. 

1.  Last  derivative  =  2.  3.  Last  derivative  =  6. 

2.  Last  derivative  =  24.  4,  Last  derivative  =  120. 

5.  Last  derivative  =  24. 


60  ALGEBRA. 


Exercise  OXLI. 

1.  1,  1,  6.  3.  -1,  -1,  -1.  5.  3,  3,  3,  -2. 

2.  2,  2,3.  4.  3,  3,-4.  6.  2,  2,  2,-6. 

7.  5,  5,  1,  -1.  9.  1.  1,  1.  -^(1=fV=II). 

8.  5,  5,  5,  -2,  -2.         10.  1,  1,  -4,  -4. 

Exercise  CXLII. 

1.  a:3  +  2ar'-18a;  +  18  =  0. 

2.  a^^+lOar'- 42a: -576  =  0. 

Z.  x^-^ci?-  1350  a:*  -  18,000  a:  +  162,000  =x  0. 

4.  a:"  +  3a:* +  24a:3-72ar'+ 1296a: -23,328  =  0. 

5.  a:*-8a:2H-4a:-224  =  0. 

Exercise  CXLIII. 

1.  3/^-83/'+ 12y  + 9  =  0.  3.  3/«  +  4y^-15y-18  =  0. 

2.  y*+2y»-82/'-6y-|-15  =  0.         4.  y»  -  5y  =  0. 

5.  y*+17y«  +  92y2  +  172y  +  96  =  0. 

6.  y*  +  4.8y3-17.86y2_56.688y  + 90.4761  =  0. 

7.  y*-f  1.2y«  +  1.54y2-  2.292y  +  3.1981  =  0. 

8.  /+3.5y*  +  4.50y3  +  5.75y^+1.8125y- 16.15625  =  0. 

9.  /-8y*  +  20y-19y2_2y  +  20  =  0. 
10.  y«  +  0.2/4-1.60y*-1.640y«  +  2.6240y2 

-  3.70208y  +  5.138944  =  0. 


Exercise  CXLV. 

1.  637. 

3. 

0.                  5.  5225.              7.  347. 

2.  -596. 

4. 

-499.          6.  82.                  8.  60146, 

9. 

-182. 

10.   11659. 

ANSWERS. 


61 


Exercise  CXLVI. 


1.  1.+,  0.4+,  -1.+. 

2.  2.+,  -0.6+,  -1.+. 

3.  -1.+,  1.+,  4.+. 

4.  1.+,  1.+,  -3.+. 

5.  -7.+.  3.+.  1.+. 


6.  1.+,  5.+,  -0.6+. 

7.  -1.+, -3.+,  -4.+. 

8.  1.+,  6.+,  7.+. 

9.  0.7+,  2.+,  -0.7+,  5.+. 
10.  2.+,  0.4+,  0.6+,  -3.+. 


1.  2.833066+. 

2.  4.264429+. 

3.  1.157451+. 

4.  2.645751+. 

5.  -3.907378+. 


Exercise  CXLVII. 

6.  8.414455+. 

7.  4.898979+. 

8.  32.865384+. 

9.  624. 
10.  29. 


11.  43. 

12.  742.826359+. 

13.  7.103575+. 

14.  -4.532088+. 

15.  -0.732050+. 


Example  in  §  622  :     3.353,  3.415. 

Exercise  OXLVIII. 
:1,  i(-7±3V5). 


2.  i(V5-l±V-10-2V5),    -i(V5+l=FV-10+2V5). 
3.  ±  1,  i(3  +  V=^rfcV-14=b6V^7), 
i  (3  -  V-^  ±  V-14-6V^. 

4.  2±V3,  i(l±V^).       6.  -1,  2zfcV3,  |(1±V^). 

5.  1,  l,i(l±  V=^T5).  7.  3±2V2,  2±V3. 
8.   -l.i(l 


m 


Vm'-2m-3). 
9.  -1,  i(l+V5±:V2V5-10),  i(l-Vb±-yj~2Vb-10). 


10.  1,  iV-l,  J(-1±V=^). 

Exercise  CXLIX. 

1.  2.438+.        5.  0.951+.        9.   1.775. 

2.  2.096+.        6.  1.900+.      10.  2.925. 

3.  2.872+.        7.  0.569+.      11.  15  years. 

4.  2.970+.        8.  0.801+.      12.   9  years. 


13.  20  years. 

14.  8  years. 

15.  17  years. 


62  ALGEBRA. 

Exercise  GL. 

1.  -2,  -5,  -5.  5.  -6,  3±5V^=^. 

2.  10,  i(ll±5V=^).  6.  8,  -4±3\/^. 

3.  2,  2±V==1.  7.  1,  1,  -2. 

4. 1,  -2=fc3V=n:.  8.  -11,  ^(iiiV^^n^), 

Exercise  CLI. 

1.  1.1542,  -0.57709  =b  1.9998 V^. 

2.  -0.41813,  0.20906  ±  2.6704  V^. 

3.  -3.2264,  1.6132  =b  0.89836  V^. 

4.  2.4567,  -1.2283  d=  0.72557  V^. 

5.  1,  1.5616,  -2.5616. 


APPENDIX. 

ExEECiSE  CXXI.  {Elements  of  Algebra). 

*    32  16  8  4  2  *^   ■ 

o     32a^5      2Qai'^5^     Sa^S*     15ae5«     135  a^S^     243Z>io 

243  27  3  8  128  1024  *      . 

3.    0}%''^  + 10  a"&i V  +  40  ai26"a;8  +  gO  a"6V2  +  80  a"6 V«  +  32  ai^SV". 
^     128  g^S^*      l\2aWy      14  a^o^V      35ai"&V      105a"6V 
2187  243  9  12  32 

_  561a}^bY  ,  1701  a}^Y  _  2187  aV 
256  2048  16384 

5.  a^Va  +  la^x  +  21a'^Va  X  x^  +  35aV  ^  s^aVa  X  x*  +  21  ax^ 

+  7VaXiC*  +  a;''. 

6.  16 ¥  - 64  J3 V2& X m  +  224 6%2 _ 224 b^VUx  m?  +  280 6^ 

'-  1126 V26  X  m»  +  56 bm^ -  SVU  x  m^  +  m\ 


ANSWERS.  63 


7.    27c3 V3 c  +  378 c^a  +  IdQcWS cxa-'  +  2520 c^a^  +  1680c\/3 c  X  a* 
+  2016ca5  +  448V37x  a^  +  128al 

It)  A  A 

-  6804  aVfa  X  y^  +  10,206  ay^  -  1 7,496  VJa  X  y'  +  6561 3/8. 

9.   64 e^  +  192  ae2\/^  +  240  a V  +  160a3ev^+  60a*e  +  12a5\/i'  +  a«. 

10    64  gg     64a5V2^     160  a^     2,%)a?xyPIx     ^^-'^-'^ 


T2^  81  27  27  3 

+  16aa;2\/2x  +  8ar». 

^j    243  g^     405a^Vj^     1350^3;     \bc?xyf\x     \bax^     x^y/\x 
1024  256  64  16  16  4     ' 

12.    64g6-  576a5v^  +  2160a*2/  -  \Z%)a^yyfy  +  4860ay 
-2916g2/2vV+729y3. 

10    ^14  ^  7g^^VI  ^  21  gio^  ^  35  g^z  VJ  ^  35  ah^  ,  21  g^z^  yfl 


16  32 


TgV     2V2 
64         128 ' 


14.  h"  -  8  W  ^hy  +  28  h^y  -  56  h'^yVb^  +  70  &y  -  56  Jy^  v^ 

+  28by^-8y^Vby  +  y\ 

15.  8c3+  24 c2 Vec^^- 180 c2a;+ 120 ca^V6cx+270ca;2+54a;2\/6cx+27x3. 

16.  8,200,192  g«.  18.    326,592 rc^^.  20.    l^^'^^"^ 

256 

17    352,716 gii(f  10.  19.   -12,033,222,8802/81.         21.   -165a%2. 

22.  (1  +  x)-^  ==l-x  +  x'^-x^+x*-a^  + 

l_l  +  i_i+l_l  + =  0.9. 

9     92      93     9*     95 

23.  {l+xf=l+^-^+^-^\^- =  1+1 -A 

2      8       16      128      256  16     512 

_27__^2^.+_^70L._ ^1.1726. 

8192      524108      8388608 

24    (l+a;)-^=l-^  +  ?JL'_5£!  +  35a^*_.63x5     =  0.998503367... 

2       8        16      128       256 


64 

ALGEBRA. 

25. 

0-5  -  5a-«a;  +  15a-7a;2  -  35a-8a;3  + 

26. 

a-«  +  6a-7a;  +  21a-8x2  +  56a-»a;3  ^ 

27. 

1     +^  +  ^1^+1^+ 

64  6«     6467     25668     64  6* 

28. 

512     92122     92160^2     67584023 
c9          cw             cii                ci2 

... 

29. 

1        9a;       45a;2      165ar« 
a'     2aw      4  a"       8  a" 

30. 

1       5a;'      15a;«     35a;» 
a^®      a"        a^*         a^* 

31. 

1        6.^^21^^  56a^^ 

«'      a^V^        «*        a*VS 

32. 

V  6  + + 

2v/6     86V6     1662V6 

33. 

^Vj         a;            3x-^               7x^ 

4:y/b^     32hW     12862^ 

34. 

.+  «        -%    ^'        

2a;     8a;'      16a;* 

35. 

• 

a       1         1          1         

2a     8  a'     16a5 

36. 

^      a     4a'     68  a' 
9       81      2187 

' 

37. 

1            1       I       3               5 

aVa     2aVa     80^^^     16oi0v'a 

38. 

1        a       3a2       5a' 
a;      2.'c'      8a;5  '  lQ^^ 

39. 

-f-^ 

xi«     14a;i5                        1      3A       SA^ 

9          81                       *   8      256      2048 

13  ;i' 

65536 

40. 

^      2d     12d^     88 d'              ^o     1    ,  ^'    ,   5a;* 
5         25         125                 "■   27     81      1458 

,    35a;6 
^  39366 

43. 

7.2801098. 

45.  2.8716219 47. 

4.073062.... 

44. 

4.431047... 

46.  3.0385047 48. 

3.87827796...- 

